Solve:

$(i)$ $x (x + 1) + (x + 2) (x + 3) = 42$

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Solution

$x (x + 1) + (x + 2) (x + 3) = 42$

$\Rightarrow x^{2} + x + x^{2} + 5 x + 6 = 42$

$\Rightarrow 2 x^{2} + 6 x - 36 = 0$

$\Rightarrow x^{2} + 3 x - 18 = 0$

$\Rightarrow x^{2} + 6 x - 3 x - 18 = 0$

$\Rightarrow x (x + 6) - 3 (x + 6) = 0$

$\Rightarrow (x + 6) (x - 3) = 0$

$\Rightarrow x + 6 = 0 or x - 3 = 0$

$\Rightarrow x = - 6 or x = 3$

$∴ x = - 6 or x = 3.$

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