Factorize $(x + y)^{2} + 9 (x + y) + 8$

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Solution

Let $x + y = p$
Then the equation is $p^{2} + 9 p + 8$
Coefficient of $p^{2} = 1$ ; Constant term = 8
Their product $= 1 \times 8 = 8$
Coefficient of p = 9
$∴$ product = 8; sum = 9
$p^{2} + 9 p + 8 = p^{2} + p + 8 p + 8$
$= p (p + 1) + 8 (p + 1)$
$= (p + 1) (p + 8)$
substituting, $p = x + y$
$∴ (x + y)^{2} + 9 (x + y) + 8 = (x + y + 1) (x + y + 8)$

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