Finding values in triple bracket question (algebra)

Question

How would I find the valued of $p$ and $q$ in the equation:

$$(x+p)(x+q)(x+5)= x^3+8x^²-3x-90$$

Any answers are well appreciated!

Answer 1

Hint: Use that $$(x+p)(x+q)(x+5)=x^3+x^2(p+q+5)+x(5p+5q+pq)+5pq$$ and compare the coefficients with $$x^3+8x^2-3x-90$$ so it must be $$8=p+q+5$$ $$-3=5p+5q+pq$$ and $$5pq=-90$$ Can you finish?

Answer 2

The problem could be simpler if we start computing$$\frac{x^3+8x^2-3x-90}{x+5}$$ Now, when done, just use the quadratic formulae for the roots of the resulting polynomial.

Answer 3

$$x^3+8x^2-3x-90 = x^3+5x^2+3x^2 -3x-90$$ $$ = x^2(x+5)+3(x^2-x-30)$$

$$ = x^2(x+5)+3(x-6)(x+5)$$ $$ = (x+5)(x^2+3x-18)$$ $$ = (x+5)(x+6)(x-3)$$

so $p= 6$ and $q=-3$