Question

Factorize:
40 + 3x − x²

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ 40+3x-x^2 \\ =-\left(x^2-3x-40\right) \end{gathered}$$

We have to split ($-$3) into two numbers such that their sum is ($-$3) and their product is ($-$40).
Clearly, $-8+5=-3\mathrm{and}-8\times 5=-40$.

$$\begin{gathered} \therefore 40+3x-x^2=-\left(x^2-3x-40\right) \\ =-\left(x^2-8x+5x-40\right) \\ =-\left[x\left(x-8\right)+5\left(x-8\right)\right] \\ =-\left(x-8\right)\left(x+5\right) \\ =\left(8-x\right)\left(5+x\right) \end{gathered}$$