Question

How do you factor the trinomial `y^2(x-7)+2y(x-7)-63(x-7)`?

Answer 1

`y^2(x-7)+2y(x-7)-63(x-7) = (x-7)(y+9)(y-7)`

Explanation:

Extracting the obvious common factor of `(x-7)`
`color(white)("XX")color(red)((x-7))color(blue)((y^2+2y-63))`

This may have been a trick question to see if you would stop at this point, but we won't.

Factoring `color(blue)((y^2+2y-63))`
requires that we find two numbers, one negative and one positive, whose product is `(-63)` and whose sum is `(+2)`

With a bit of consideration we come up with the pair `(-7,+9)`
So
`color(white)("XX")color(blue)((y^2+2y-63)) = color(blue)((y-7)(y+9))`

and the complete original expression can be factored as
`color(white)("XX")color(red)((x-7)color(blue)((y-7)(y+9))`