Question

How do you factor the expression `10x^2+20x-80`?

Answer 1

You factor a polynomial by finding (when possible) its roots `x_i` and dividing the polynomial by the linear factor `(x-x_i)`.

In your case, first of all, let's factor `10` out of the polynomial, obtaining
`10x^2+20x-80 = 10(x^2+2x-8)`

Without bothering with the discriminant formula, we can solve `x^2+2x-8` by remembering the result that, when a quadratic polynomial is of the form `x^2-sx+p`, the sum of the solutions is `s`, and the product is `p`. In this case, we need to find two numbers `x_1` and `x_2` such that `x_1+x_2=-2`, and `x_1x_2=-8`.

It should be easy to find that the two numbers are `-4` and `2`. So, the linear factors are `(x+4)(x-2)`. The factorization of your polynomial is thus `10(x+4)(x-2)`.