Quadratic in product form

Statement

Solve the equation $x(x+7) = 6(-6 - x)$.

Solution

Expand both sides: $x^2 + 7x = -36 - 6x.$ Move everything to the left: $x^2 + 7x + 6x + 36 = 0 \implies x^2 + 13x + 36 = 0.$ Look for two numbers with sum $13$ and product $36$: they are $4$ and $9$. Hence $(x+4)(x+9) = 0$, so $x = -4$ or $x = -9$. Check: for $x=-4$: $(-4)(3) = -12$ and $6(-6+4) = -12$; for $x=-9$: $(-9)(-2) = 18$ and $6(-6+9) = 18$. Correct. $\boxed{x = -4 \ \vee\ x = -9}$