Factorise $(x^{2} + x)^{2} + 4 (x^{2} + x) - 12$ .

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Solution

Given expression is $(x^{2} + x)^{2} + 4 (x^{2} + x) - 12$ .
We can write the given expression as $(x^{2} + x)^{2} + 4 (x^{2} + x) + 4 - 16$
Now, using the property, $a^{2} + 2 a b + b^{2} = (a + b)^{2}$
with $a =$ $(x^{2} + x)$ , $b = 2$ , we can write the expression as,
$(x^{2} + x + 2)^{2} - 16$
$\Rightarrow (x^{2} + x + 2)^{2} - (4)^{2}$
Using $a^{2} - b^{2} = (a + b) (a - b)$ , we can write the expression as,
$(x^{2} + x + 2 - 4) (x^{2} + x + 2 + 4)$
$\Rightarrow (x^{2} + x - 2) (x^{2} + x + 6)$

Hence, $(x^{2} + x)^{2} + 4 (x^{2} + x) - 12 = (x^{2} + x - 2) (x^{2} + x + 6)$ .

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