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Simplification or other simple results

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4a^3+12a^2+5a

This solution deals with simplification or other simple results.

Solution found

a*(2a+1)*(2a+5)
a*(2a+1)*(2a+5)

Step by Step Solution

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Step  1  :

Equation at the end of step  1  :

  ((4 • (a3)) +  (22•3a2)) +  5a

Step  2  :

Equation at the end of step  2  :

  (22a3 +  (22•3a2)) +  5a

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4a3 + 12a2 + 5a  =   a • (4a2 + 12a + 5) 

Trying to factor by splitting the middle term

 4.2     Factoring  4a2 + 12a + 5 

The first term is,  4a2  its coefficient is  4 .
The middle term is,  +12a  its coefficient is  12 .
The last term, "the constant", is  +5 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 5 = 20 

Step-2 : Find two factors of  20  whose sum equals the coefficient of the middle term, which is   12 .

     -20   +   -1   =   -21
     -10   +   -2   =   -12
     -5   +   -4   =   -9
     -4   +   -5   =   -9
     -2   +   -10   =   -12
     -1   +   -20   =   -21
     1   +   20   =   21
     2   +   10   =   12   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  2  and  10 
                     4a2 + 2a + 10a + 5

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

Final result :

  a • (2a + 1) • (2a + 5)

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