Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (3x2 -  24x) +  45

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3x² - 24x + 45  =   3 • (x² - 8x + 15) 

Trying to factor by splitting the middle term

 3.2     Factoring  x² - 8x + 15 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -8x  its coefficient is  -8 .
The last term, "the constant", is  +15 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 15 = 15 

Step-2 : Find two factors of  15  whose sum equals the coefficient of the middle term, which is   -8 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -5  and  -3 
                     x² - 5x - 3x - 15

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-5)
              Add up the last 2 terms, pulling out common factors :
                    3 • (x-5)
Step-5 : Add up the four terms of step 4 :
                    (x-3)  •  (x-5)
             Which is the desired factorization

Final result :

  3 • (x - 3) • (x - 5)