Solve the quadratic equation, x^2 - 4x -5 = 0

Solve x²- 4x - 5 = 0
A very full explanation would be:
Start by listing the pairs of numbers that can be multiplied together to get 1 * -5 = -5 (i.e. the coefficient of the first term multiplied by the last term) . [5 is prime so] the only options are -5 * 1 and -1 * 5. The correct pair, if this can be factorised, will be the pair that contains numbers which, when added together, will equal the coefficient of the second term, here -4. Therefore the correct pair is -5 and 1. But why do we want these two numbers?
Well, in general, the fully-factorised form of a quadratic equation like this would look like (x+a)(x+b)=0. When the brackets are expanded, we find that x² + ax + bx + ab = x² + (a+ b)x + ab = 0. Comparing the terms in this equation to the specific one in the question, in our specific case we can see that a+b = -4, and ab = -5, and we just have to find numbers a and b, which we did in the first paragraph. So now we know that in this case, (x-5)(x+1)=0. From here it is simple to find the solutions, since one of the sets of brackets must equal zero if the right hand side is to equal zero (i.e. the only number that when multiplied by (x+anything) is equal to 0, is 0.) So x - 5 = 0 or x + 1 = 0. Rearanging, we can see that x = 5 or x = -1.