Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (7x2 +  8x) +  1

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  7x²+8x+1 

The first term is,  7x²  its coefficient is  7 .
The middle term is,  +8x  its coefficient is  8 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   7 • 1 = 7 

Step-2 : Find two factors of  7  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,  1  and  7 
                     7x² + 1x + 7x + 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (7x+1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (7x+1)
Step-5 : Add up the four terms of step 4 :
                    (x+1)  •  (7x+1)
             Which is the desired factorization

Final result :

  (7x + 1) • (x + 1)