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)#.