Question

How do you factor `8x^3+28x^2+24x`?

Answer 1

`8x^3+28x^2+24x = 4x(2x^2+7x+6) = 4x(2x+3)(x+2)`.

To find this, first note that all the terms are divisible by `4` and by `x`. Dividing through by `4x` yields the quadratic `2x^2+7x+6`.

If this has linear factors with integer coefficients they must be of the form `(2x+a)` and `(x+b)` in order that the product starts with `2x^2`, where `ab=6` and `2b+a=7`. From this, it's easy to see that `a=3` and `b=2`, so `2x^2+7x+6 = (2x+3)(x+2)`.