Its required by the logic of factoring and factoring the quadratic is the undo of the original binomial multiplication. We can call these values a and b and we use them in the factorization x - a x - b. When factoring a quadratic expression we find the values of x that would make the expression equal to zero.
If you dont have a calculator then I dont know of any definite way in which you can tell if a polynomial is factorable or not but you should just try playing with the numbers factor out stuff.
As the name suggests the method reduces a second degree polynomial ax2 bx c 0 into a product of simple first degree equations as illustrated in the following example. Or if your equation factored then you can use the quadratic formula to test if your solutions of the quadratic equation are correct. While factoring may not always be successful the Quadratic Formula can always find the solution. We can call these values a and b and we use them in the factorization x - a x - b.