Nomenclature: the terminology for a specific topic
Quadratic: a polynomial whose highest power is 2
Coefficient: the number before the variable

The Technique:
Let us take a concrete example: \(x^2-3x-18\). To factorise this monic quadratic we need to find two numbers that multiply to give -18 and add to give -3. Such numbers will be -6 and 3. Thus we can write \(x^{2}-3 x-18=(x+3)(x-6)\)

Examples:
Factorise:
\(x^{2}+8 x+16\)
We are looking for two numbers that multiply to give 16 and add to give 8. The numbers 4 and 4 satisfy both conditions.
Thus \(x^{2}+8 x+16=(x+4)(x+4)\)
\[
=(x+4)^{2}
\]

\(x^{2}+5 x+6\)
We are looking for two numbers that multiply to give 6 and add to give 5. The numbers 2 and 3 satisfy both conditions.
Thus \(x^{2}+5 x+6=(x+2)(x+3)\)

\(x^{2}+7 x+10\)
We are looking for two numbers that multiply to give 10 and add to give 7. The numbers 2 and 5 satisfy both conditions.
Thus \(x^{2}+7 x+10=(x+2)(x+5)\)

\(x^{2}+9 x+14\)
We are looking for two numbers that multiply to give 14 and add to give 9. The numbers 2 and 7 satisfy both conditions.
Thus \(x^{2}+9 x+14=(x+2)(x+7)\)

Exercises:
Factorise:
\(x^{2}+9 x+20\)
We are looking for two numbers that multiply to give 20 and add to give 9. The numbers 4 and 5 satisfy both conditions.
Thus \(x^{2}+9 x+20=(x+4)(x+5)\)

\(x^{2}+12 x+32\)
We are looking for two numbers that multiply to give 32 and add to give 12. The numbers 4 and 8 satisfy both conditions.
Thus \(x^{2}+12 x+32=(x+4)(x+8)\)

\(x^{2}+20 x+75\)
We are looking for two numbers that multiply to give 75 and add to give 20. The numbers 5 and 15 satisfy both conditions.
Thus \(x^{2}+20 x+75=(x+5)(x+15)\)

\(x^{2}+15 x+56\)
We are looking for two numbers that multiply to give 56 and add to give 15. The numbers 7 and 8 satisfy both conditions.
Thus \(x^{2}+15 x+56=(x+7)(x+8)\)

\(x^{2}-5 x+6\)
We are looking for two numbers that multiply to give 6 and add to give -5. The numbers -2 and -3 satisfy both conditions.
Thus \(x^{2}-5 x+6=(x-2)(x-3)\)

\(x^{2}-17 x+30\)
We are looking for two numbers that multiply to give 30 and add to give -17. The numbers -15 and -2 satisfy both conditions.
Thus \(x^{2}-17 x+30=(x-15)(x-2)\)

\(x^{2}-9 x+14\)
We are looking for two numbers that multiply to give 14 and add to give -9. The numbers -7 and -2 satisfy both conditions.
Thus \(x^{2}-9 x+14=(x-7)(x-2)\)

\(x^{2}-15 x+44\)
We are looking for two numbers that multiply to give 44 and add to give -15. The numbers -11 and -4 satisfy both conditions.
Thus \(x^{2}-15 x+44=(x-11)(x-4)\)

\(x^{2}-18 x+80\)
We are looking for two numbers that multiply to give 80 and add to give -18. The numbers -10 and -8 satisfy both conditions.
Thus \(x^{2}-18 x+80=(x-10)(x-8)\)

\(x^{2}-14 x+40\)
We are looking for two numbers that multiply to give 40 and add to give -14. The numbers -10 and -4 satisfy both conditions.
Thus \(x^{2}-14 x+40=(x-10)(x-4)\)

\(x^{2}-30 x+56\)
We are looking for two numbers that multiply to give 56 and add to give -30. The numbers -28 and -2 satisfy both conditions.
Thus \(x^{2}-30 x+56=(x-28)(x-2)\)

\(x^{2}+x-6\)
We are looking for two numbers that multiply to give -6 and add to give 1. The numbers 3 and -2 satisfy both conditions.
Thus \(x^{2}+x-6=(x-2)(x+3)\)

\(x^{2}+x-30\)
We are looking for two numbers that multiply to give -30 and add to give 1. The numbers 6 and -5 satisfy both conditions.
Thus \(x^{2}+x-30=(x-5)(x+6)\)

\(x^{2}-5 x-14\)
We are looking for two numbers that multiply to give -14 and add to give -5. The numbers -7 and 2 satisfy both conditions.
Thus \(x^{2}-5 x-14=(x+2)(x-7)\)

\(x^{2}-7 x-44\)
We are looking for two numbers that multiply to give -44 and add to give -7. The numbers -11 and 4 satisfy both conditions.
Thus \(x^{2}-7 x-44=(x+4)(x-11)\)

\(x^{2}+2 x-80\)
We are looking for two numbers that multiply to give -80 and add to give 2. The numbers 10 and -8 satisfy both conditions.
Thus \(x^{2}+2 x-80=(x-8)(x+10)\)

\(x^{2}+3 x-40\)
We are looking for two numbers that multiply to give -40 and add to give 3. The numbers 8 and -5 satisfy both conditions.
Thus \(x^{2}+3 x-40=(x-5)(x+8)\)

\(x^{2}+4 x-21\)
We are looking for two numbers that multiply to give -21 and add to give 4. The numbers 7 and -3 satisfy both conditions.
Thus \(x^{2}+4 x-21=(x-3)(x+7)\)

\(x^{2}-x+56\)
This quadratic cannot be factored over the integers because there are no two numbers that multiply to give 56 and add to give -1.

\(x^{2}-3 x+2\)
We are looking for two numbers that multiply to give 2 and add to give -3. The numbers -1 and -2 satisfy both conditions.
Thus \(x^{2}-3 x+2=(x-1)(x-2)\)

\(x^{2}-3 x-10\)
We are looking for two numbers that multiply to give -10 and add to give -3. The numbers -5 and 2 satisfy both conditions.
Thus \(x^{2}-3 x-10=(x+2)(x-5)\)

\(x^{2}-5 x-14\)
We are looking for two numbers that multiply to give -14 and add to give -5. The numbers -7 and 2 satisfy both conditions