If $(x + a) (x + b) = x^{2} - 5 x - 300$ , find the values of $a^{2} + b^{2}$

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Solution

$x^{2} - 5 x - 300 = x^{2} - 20 x + 15 x - 300 = x (x - 20) + 15 (x - 20) = (x - 20) (x + 15)$
Given $(x + a) (x + b) = x^{2} - 5 x - 300$
Then $(x + a) (x + b) = (x - 20) (x + 15)$
So, compare this we get $a = - 20$ and $b = 15$
Then $a^{2} + b^{2} = (- 20)^{2} + (15)^{2} = 400 + 225 = 625$

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