Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (5t2 -  12t) -  81

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  5t²-12t-81 

The first term is,  5t²  its coefficient is  5 .
The middle term is,  -12t  its coefficient is  -12 .
The last term, "the constant", is  -81 

Step-1 : Multiply the coefficient of the first term by the constant   5 • -81 = -405 

Step-2 : Find two factors of  -405  whose sum equals the coefficient of the middle term, which is   -12 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -27  and  15 
                     5t² - 27t + 15t - 81

Step-4 : Add up the first 2 terms, pulling out like factors :
                    t • (5t-27)
              Add up the last 2 terms, pulling out common factors :
                    3 • (5t-27)
Step-5 : Add up the four terms of step 4 :
                    (t+3)  •  (5t-27)
             Which is the desired factorization

Final result :

  (5t - 27) • (t + 3)