I wanted to know how you would know that -4x^2 + 2x +90 / x-5 would be able to factor down to (x-5)(-4x-18)/x-5

Question

By looking at -4x^2 + 2x +90 / x-5 , I would never think that it could be factored down to (x-5)(-4x-18)/x-5

How should I approach it so I know that it could be factored down to (x-5)(-4x-18)/x-5?

Answer 1

First you take out common factors.
-4x^2 + 2x +90 = -2(2x²-x-45)

Is it possible to find a, b such that ab=-45, 2a+b=-1 or a+2b=-1?

Look at the factors of the constant term, namely 45=5*9. Since 2*5-9=1, we can try the factors 5 and 9 in different ways to get:
(2x+9)(x-5)

Answer 2

rather hard to explain in detail here.

this page
http://www.recitfga.qc.ca/english/activities/sitsat-2006/jean-foster/0-3.htm
gives a reasonable explanation.

I became suspicious when I saw the simple factor x-5 in the denominator, and had a sneaking suspicion that it could also be in the top.
so I subbed in x=5 into the top and sure enough, I got a zero for the result.
(recall the factor theorem, which says that if f(a) = 0 then x-a is a factor)

so I know that
-4x^2 + 2x +90
= (x - 5)(?x ± ?)
now a bit of logic,
what multiplied by x would give me -4x^2 ? , clearly -4x
what multiplied by -5 would give me +90 ? clearly -18
so
-4x^2 + 2x +90 = (x-5)(-4x-18)

BTW, the answer can be taken further by dividing out the x-5 and simplifying the top
(x-5)(-4x-18)/x-5
= -2(2x + 9), x cannot be equal to 5