The quadratics for this section are a little less tame. The $x^2$ term now possesses a coefficient which we will be trying to pull out of the entire expression to use our techniques from section 4 on the new expression within the parenthesis.


\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \question[3] Factorise:
        \begin{parts}
            \part \(3 x^{2}+9 x+6\)
            \begin{solutionordottedlines}[1in]
                \(3 x^{2}+9 x+6=3\left(x^{2}+3 x+2\right) \quad\) (Take out the common factor.)

                \[
                =3(x+2)(x+1)
                \]
            \end{solutionordottedlines}
            \part \(6 x^{2}-54\)
            \begin{solutionordottedlines}[1in]
                \(6 x^{2}-54=6\left(x^{2}-9\right)\)

                \[
                =6(x+3)(x-3)
                \]
            \end{solutionordottedlines}
            \part \(-x^{2}-x+2\)
            \begin{solutionordottedlines}[1in]
                \(-x^{2}-x+2=-\left(x^{2}+x-2\right)\)

                (Factor -1 from each term.)

                \[
                =-(x+2)(x-1)
                \]
            \end{solutionordottedlines}
        \end{parts}
        \question[10]
        \begin{parts}\begin{multicols}{2}
            \part \(2 x^{2}+14 x+24\)
            \begin{solutionordottedlines}[1in]
                \(2 x^{2}+14 x+24=2(x^{2}+7 x+12)\)

                \[
                =2(x+3)(x+4)
                \]
            \end{solutionordottedlines}
            \part \(4 x^{2}-24 x+36\)
            \begin{solutionordottedlines}[1in]
                \(4 x^{2}-24 x+36=4(x^{2}-6 x+9)\)

                \[
                =4(x-3)^{2}
                \]
            \end{solutionordottedlines}
            \part \(4 x^{2}-4 x+48\)
            \begin{solutionordottedlines}[1in]
                \(4 x^{2}-4 x+48=4(x^{2}-x+12)\)

                (This quadratic does not factor over the integers.)
            \end{solutionordottedlines}
            \part \(3 x^{2}+9 x-120\)
            \begin{solutionordottedlines}[1in]
                \(3 x^{2}+9 x-120=3(x^{2}+3 x-40)\)

                \[
                =3(x+8)(x-5)
                \]
            \end{solutionordottedlines}
            \part \(2 x^{2}-4 x-96\)
            \begin{solutionordottedlines}[1in]
                \(2 x^{2}-4 x-96=2(x^{2}-2 x-48)\)

                \[
                =2(x-8)(x+6)
                \]
            \end{solutionordottedlines}
            \part \(3 x^{2}-18 x+27\)
            \begin{solutionordottedlines}[1in]
                \(3 x^{2}-18 x+27=3(x^{2}-6 x+9)\)

                \[
                =3(x-3)^{2}
                \]
            \end{solutionordottedlines}
            \part \(4 x^{2}-16\)
            \begin{solutionordottedlines}[1in]
                \(4 x^{2}-16=4(x^{2}-4)\)

                \[
                =4(x+2)(x-2)
                \]
            \end{solutionordottedlines}
            \part \(3 a^{2}-27\)
            \begin{solutionordottedlines}[1in]
                \(3 a^{2}-27=3(a^{2}-9)\)

                \[
                =3(a+3)(a-3)
                \]
            \end{solutionordottedlines}
            \part \(27 x^{2}-3 y^{2}\)
            \begin{solutionordottedlines}[1in]
                \(27 x^{2}-3 y^{2}=3(9x^{2}-y^{2})\)

                \[
                =3(3x+y)(3x-y)
                \]
            \end{solutionordottedlines}
            \part \(128-2 x^{2}\)
            \begin{solutionordottedlines}[1in]
                \(128-2 x^{2}=-2(x^{2}-64)\)

                \[
                =-2(x+8)(x-8)
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \begin{parts}\begin{multicols}{2}
            \setcounter{partno}{10}
            \part \(\frac{1}{4} a^{2}-9\)
            \begin{solutionordottedlines}[1in]
                \(\frac{1}{4} a^{2}-9=\left(\frac{1}{2}a\right)^{2}-3^{2}\)

                \[
                =\left(\frac{1}{2}a+3\right)\left(\frac{1}{2}a-3\right)
                \]
            \end{solutionordottedlines}
            \part \(-x^{2}-8 x-12\)
            \begin{solutionordottedlines}[1in]
                \(-x^{2}-8 x-12=-1(x^{2}+8 x+12)\)

                \[
                =-(x+6)(x+2)
                \]
            \end{solutionordottedlines}
            \part \(9+8 x-x^{2}\)
            \begin{solutionordottedlines}[1in]
                \(9+8 x-x^{2}=-(x^{2}-8 x-9)\)

                \[
                =-(x-9)(x+1)
                \]
            \end{solutionordottedlines}
            \part \(-x^{2}+3 x+40\)
            \begin{solutionordottedlines}[1in]
                \(-x^{2}+3 x+40=-1(x^{2}-3 x-40)\)

                \[
                =-(x-8)(x+5)
                \]
            \end{solutionordottedlines}
            \part \(11 x-x^{2}-24\)
            \begin{solutionordottedlines}[1in]
                \(11 x-x^{2}-24=-(x^{2}-11 x+24)\)

                \[
                =-(x-8)(x-3)
                \]
            \end{solutionordottedlines}
            \part \(-16 x-63-x^{2}\)
            \begin{solutionordottedlines}[1in]
                \(-16 x-63-x^{2}=-(x^{2}+16 x+63)\)

                \[
                =-(x+7)(x+9)
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
    \end{questions}
\end{examplebox}

\begin{exercisebox}
    \subsection{Exercises:}
    \begin{questions}
        \question[16] Factorise:
        \begin{parts}\begin{multicols}{2}
            \part \(3 x^{2}+24 x+36\)
            \begin{solutionordottedlines}[1in]
                \(3 x^{2}+24 x+36=3(x^{2}+8 x+12)\)

                \[
                =3(x+6)(x+2)
                \]
            \end{solutionordottedlines}
            \part \(7 x^{2}+14 x+7\)
            \begin{solutionordottedlines}[1in]
                \(7 x^{2}+14 x+7=7(x^{2}+2 x+1)\)

                \[
                =7(x+1)^{2}
                \]
            \end{solutionordottedlines}
            \part \(2 x^{2}-18 x+36\)
            \begin{solutionordottedlines}[1in]
                \(2 x^{2}-18 x+36=2(x^{2}-9 x+18)\)

                \[
                =2(x-6)(x-3)
                \]
            \end{solutionordottedlines}
            \part \(3 x^{2}-3 x-90\)
            \begin{solutionordottedlines}[1in]
                \(3 x^{2}-3 x-90=3(x^{2}-x-30)\)

                \[
                =3(x-6)(x+5)
                \]
            \end{solutionordottedlines}
            \part \(5 x^{2}+65 x+180\)
            \begin{solutionordottedlines}[1in]
                \(5 x^{2}+65 x+180=5(x^{2}+13 x+36)\)

                \[
                =5(x+9)(x+4)
                \]
            \end{solutionordottedlines}
            \part \(5 x^{2}-20 x+20\)
            \begin{solutionordottedlines}[1in]
                \(5 x^{2}-20 x+20=5(x^{2}-4 x+4)\)

                \[
                =5(x-2)^{2}
                \]
            \end{solutionordottedlines}
            \part \(2 x^{2}-18\)
            \begin{solutionordottedlines}[1in]
                \(2 x^{2}-18=2(x^{2}-9)\)

                \[
                =2(x+3)(x-3)
                \]
            \end{solutionordottedlines}
            \part \(6 x^{2}-600\)
            \begin{solutionordottedlines}[1in]
                \(6 x^{2}-600=6(x^{2}-100)\)

                \[
                =6(x+10)(x-10)
                \]
            \end{solutionordottedlines}
            \part \(45-5 b^{2}\)
            \begin{solutionordottedlines}[1in]
                \(45-5 b^{2}=-5(b^{2}-9)\)

                \[
                =-5(b+3)(b-3)
                \]
            \end{solutionordottedlines}
            \part \(\frac{1}{2} a^{2}-2 b^{2}\)
            \begin{solutionordottedlines}[1in]
                \(\frac{1}{2} a^{2}-2 b^{2}=\frac{1}{2}(a^{2}-4 b^{2})\)

                \[
                =\frac{1}{2}(a+2b)(a-2b)
                \]
            \end{solutionordottedlines}
            \part \(\frac{1}{5} x^{2}-20\)
            \begin{solutionordottedlines}[1in]
                \(\frac{1}{5} x^{2}-20=\frac{1}{5}(x^{2}-100)\)

                \[
                =\frac{1}{5}(x+10)(x-10)
                \]
            \end{solutionordottedlines}
            \part \(12-11 x-x^{2}\)
            \begin{solutionordottedlines}[1in]
                \(12-11 x-x^{2}=-(x^{2}+11 x-12)\)

                \[
                =-(x-1)(x+12)
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \begin{parts}\begin{multicols}{2}
            \part \(-x^{2}-4 x-4\)
            \begin{solutionordottedlines}[1in]
                \(-x^{2}-4 x-4=-1(x^{2}+4 x+4)\)

                \[
                =-(x+2)^{2}
                \]
            \end{solutionordottedlines}
            \part \(42+x-x^{2}\)
            \begin{solutionordottedlines}[1in]
                \(42+x-x^{2}=-(x^{2}-x-42)\)

                \[
                =-(x-7)(x+6)
                \]
            \end{solutionordottedlines}
            \part \(-3 x^{2}-30 x+72\)
            \begin{solutionordottedlines}[1in]
                \(-3 x^{2}-30 x+72=-3(x^{2}+10 x-24)\)

                \[
                =-3(x-2)(x+12)
                \]
            \end{solutionordottedlines}
            \part \(-x^{2}-35+12 x\)
            \begin{solutionordottedlines}[1in]
                \(-x^{2}-35+12 x=-1(x^{2}-12 x+35)\)

                \[
                =-(x-7)(x-5)
                \]
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
    \end{questions}

\end{exercisebox}
