Solution - Simplification or other simple results

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  a²+4a-45 

The first term is,  a²  its coefficient is  1 .
The middle term is,  +4a  its coefficient is  4 .
The last term, "the constant", is  -45 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -45 = -45 

Step-2 : Find two factors of  -45  whose sum equals the coefficient of the middle term, which is   4 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -5  and  9 
                     a² - 5a + 9a - 45

Step-4 : Add up the first 2 terms, pulling out like factors :
                    a • (a-5)
              Add up the last 2 terms, pulling out common factors :
                    9 • (a-5)
Step-5 : Add up the four terms of step 4 :
                    (a+9)  •  (a-5)
             Which is the desired factorization

Final result :

  (a + 9) • (a - 5)