Solution - Simplification or other simple results

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²+6x+8 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +6x  its coefficient is  6 .
The last term, "the constant", is  +8 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 8 = 8 

Step-2 : Find two factors of  8  whose sum equals the coefficient of the middle term, which is   6 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  2  and  4 
                     x² + 2x + 4x + 8

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

Equation at the end of step  1  :

  x +  4 • (x + 4) • (x + 2)

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  4x²+25x+32 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  +25x  its coefficient is  25 .
The last term, "the constant", is  +32 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 32 = 128 

Step-2 : Find two factors of  128  whose sum equals the coefficient of the middle term, which is   25 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  4x2 + 25x + 32