Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (3t2 +  30t) +  63

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3t² + 30t + 63  =   3 • (t² + 10t + 21) 

Trying to factor by splitting the middle term

 3.2     Factoring  t² + 10t + 21 

The first term is,  t²  its coefficient is  1 .
The middle term is,  +10t  its coefficient is  10 .
The last term, "the constant", is  +21 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 21 = 21 

Step-2 : Find two factors of  21  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,  3  and  7 
                     t² + 3t + 7t + 21

Step-4 : Add up the first 2 terms, pulling out like factors :
                    t • (t+3)
              Add up the last 2 terms, pulling out common factors :
                    7 • (t+3)
Step-5 : Add up the four terms of step 4 :
                    (t+7)  •  (t+3)
             Which is the desired factorization

Final result :

  3 • (t + 7) • (t + 3)