Simplify
$12 - (x + x^{2}) (8 - x - x^{2})$

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Solution

$12 - (x + x^{2}) (8 - x - x^{2})$
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$(x + x^{2}) (8 - x - x^{2})$

$= x \cdot 8 + x (- x) + x (- x^{2}) + x^{2} \cdot 8 + x^{2} (- x) + x^{2} (- x^{2})$

$= 8 x - x x - x^{2} x + 8 x^{2} - x^{2} x - x^{2} x^{2}$

$= - x^{4} - 2 x^{3} + 7 x^{2} + 8 x$

$= - (- x^{4} - 2 x^{3} + 7 x^{2} + 8 x)$

$= x^{4} + 2 x^{3} - 7 x^{2} - 8 x$
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$= 12 + x^{4} + 2 x^{3} - 7 x^{2} - 8 x$

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