Factorise the expression into factors by expressing them as the difference of two squares: $x^{2} + 5 x + 6$ [2 marks]

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Solution

$x^{2} + 5 x + 6$
$= x^{2} + 2 \times x \times \frac{5}{2} + (\frac{5}{2})^{2} - (\frac{5}{2})^{2} + 6$
$= (x + \frac{5}{2})^{2} - \frac{25}{4} + 6$ [1 mark]
$= (x + \frac{5}{2})^{2} - \frac{1}{4}$
$= (x + \frac{5}{2})^{2} -$ $(\frac{1}{2})^{2}$
$= (x + \frac{5}{2} - \frac{1}{2})$ $(x + \frac{5}{2} + \frac{1}{2})$ $[s i n c e, a^{2} - b^{2} = (a - b) (a + b)]$
$= (x + 2) (x + 3)$ [1 mark]

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