The reason why we factorise quadratic equation is because it is a neater and simpler way of writing the expression. 

Example 1

If we wanted to factorise x^2 + 10 + 21 we must look at factors of 21 that add up to 10. 

The factors are: 

7 and 3

21 and 1 

It is clear that the factors 7 and 3 add up to 10, thus they are the factors that we shall use. 

Therefore we can write (x+7)(x+3) = x^2 + 10x + 21

Example 2 

If the question has the form of x^2 - bx + c such as x^2 - 10x +21, we use the same technique as example one and thus the factors are -7 and -3 (note they add up to -10). 

Finally this means x^2 - 10x + 21 = (x-7)(x-3)

Example 3

If the equation has the form x^2 + bx - c such as x^2 + 4x - 21. 

As above we find the factors of 21 BUT now we find the difference of the two factors to obtain 4.

The rule tends to be if the middle number is positive the larger factor is positive and if the middle number is negative the larger factor is negative.

(x+7)(x-3) = x^2 + 4x - 21

Also note:

(x-7)(x+3) = x^2 - 4x - 21