Question

How do you factor `x^3 - 4x^2 -7x = -10`?

Answer 1

`(x-1)(x-5)(x+2)=0`

Explanation:

Re-writing `x^3-4x^2-7x=-10` as
`x^3-4x^2-7x+10=0`

We notice that the sum of the coefficients is zero
`rArr x=1` is a solution to this equation
`rArr (x-1)` is a factor of `x^3-4x-7x+10`

Dividing `x^3-4x-7x+10` by `(x-1)` by polynomial long division or synthetic division gives:
`color(white)("XXX")(x-1)(x^2-3x-10)`

If there is a factoring of `(x^2-3-10)` with integer coefficients
then we need to find a pair of factors of `(-10)` which add up to `(-3)`
With a bit of thought we can come up with `(-5,2)`
which gives us the complete factoring:
`color(white)("XXX")(x-1)((x-5)(x+2)`