Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (3t2 -  30t) +  72

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3t² - 30t + 72  =   3 • (t² - 10t + 24) 

Trying to factor by splitting the middle term

 3.2     Factoring  t² - 10t + 24 

The first term is,  t²  its coefficient is  1 .
The middle term is,  -10t  its coefficient is  -10 .
The last term, "the constant", is  +24 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 24 = 24 

Step-2 : Find two factors of  24  whose sum equals the coefficient of the middle term, which is   -10 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  -4 
                     t² - 6t - 4t - 24

Step-4 : Add up the first 2 terms, pulling out like factors :
                    t • (t-6)
              Add up the last 2 terms, pulling out common factors :
                    4 • (t-6)
Step-5 : Add up the four terms of step 4 :
                    (t-4)  •  (t-6)
             Which is the desired factorization

Final result :

  3 • (t - 4) • (t - 6)