Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (22p2 +  4p) +  1

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  4p²+4p+1 

The first term is,  4p²  its coefficient is  4 .
The middle term is,  +4p  its coefficient is  4 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   4 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  2  and  2 
                     4p² + 2p + 2p + 1

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

Multiplying Exponential Expressions :

 2.2    Multiply  (2p+1)  by  (2p+1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (2p+1)  and the exponents are :
          1 , as  (2p+1)  is the same number as  (2p+1)¹ 
 and   1 , as  (2p+1)  is the same number as  (2p+1)¹ 
The product is therefore,  (2p+1)⁽¹⁺¹⁾ = (2p+1)² 

Final result :

  (2p + 1)2