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In a quadratic expression, the highest power of \(x\) is \(x^2\).

A quadratic expression can sometimes be factorised into two brackets in the form of \((x + a)(x + b)\) where \(a\) and \(b\) can be any term, positive, negative or zero. \(a\) and \(b\) can be found by using a product and sum method.

Expanding the brackets \((x + 2)(x + 3)\) gives \(x^2 + 3x + 2x + 6\)

Which simplifies to: \(x^2 + 5x + 6\)

Factorising is the reverse process of expanding brackets, so factorising \(x^2 + 5x + 6\) gives \((x + 2)(x + 3)\).

Factorise \(x^2 + 7x + 10\).

There are a couple of ways of making +10 by multiplying two numbers. These are \(1 \times 10\) and \(2 \times 5\). Only the combination of 2 and 5 will also give a sum of +7, so the two numbers are 2 and 5.

\(x^2 + 7x + 10 = (x + 2)(x + 5)\)

\((x + 2)(x + 5) = x \times x + x \times 5 + 2 \times x + 2 \times 5\)

\(= x^2 + 5x + 2x + 10 = x^2 + 7x + 10\)

This is the original expression, so \((x + 2)(x + 5)\) is the correct factorisation.

Factorise \(y^2 - 12y + 11\).

11 is a prime number, so the only way of multiplying two numbers to make 11 is \(11 \times 1\).

\(11 + 1\) will add up to +12, not -12, so the numbers must be -11 and -1 as \(-11 \times -1 = +11\).

\(= (y - 11)(y - 1)\)

\((y - 11)(y - 1) = (y \times y) + (y \times -1) + (-11 \times y) + (-11 \times -1)\)

\(= y^2 - y - 11y + 11\) which simplifies to \(y^2 - 12y + 11\).

This is the original expression, so \((y - 11)(y - 1)\) is the correct factorisation.

The difference of two numbers is found by subtracting. The difference of two squares means one squared term subtract another squared term. For example, \(x^2 - 9\) would be the difference of two squares as \(x^2\) is a squared term (\(x\) has been multiplied by itself) and 9 is a square number (\(3 \times 3\)).

Factorise \(x^2 - 4\).

Quadratics can be factorised into the form \((x + a)(x + b)\).

\(x^2 - 4\) can be written as \(x^2 + 0x -4\).

The factor pairs that make -4 are either \(-1 \times 4\), \(1 \times -4\) or \(-2 \times 2\). The factor pair that has a product of -4 but a sum of 0 is \(-2 \times 2\) because \(-2 \times 2 = -4\) and \(-2 + 2 = 0\).

This gives \((x - 2)(x + 2)\).

\((x - 2)(x + 2) = (x \times x) + (x \times 2) + (-2 \times x) + (-2 \times 2)\)

\(= x^2 + 2x - 2x - 4\). Collecting the like terms gives the original expression of \(x^2 - 4\).