Solution - Factoring binomials using the difference of squares

Step  1  :

             1
 Simplify   ——
            64

Equation at the end of step  1  :

                   1
  (125 • (a3)) +  ——
                  64
 Step  2  :

Equation at the end of step  2  :

           1
  53a3 +  ——
          64

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  64  as the denominator :

             53a3     53a3 • 64
     53a3 =  ————  =  —————————
              1          64    

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 53a3 • 64 + 1     8000a3 + 1
 —————————————  =  ——————————
      64               64    

Trying to factor as a Sum of Cubes :

 3.3      Factoring:  8000a³ + 1 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  8000  is the cube of  20 

Check :  1  is the cube of   1 
Check :  a³ is the cube of   a¹

Factorization is :
             (20a + 1)  •  (400a² - 20a + 1) 

Trying to factor by splitting the middle term

 3.4     Factoring  400a² - 20a + 1 

The first term is,  400a²  its coefficient is  400 .
The middle term is,  -20a  its coefficient is  -20 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   400 • 1 = 400 

Step-2 : Find two factors of  400  whose sum equals the coefficient of the middle term, which is   -20 .

For tidiness, printing of 24 lines which failed to find two such factors, was suppressed

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

Final result :

  (20a + 1) • (400a2 - 20a + 1)
  —————————————————————————————
               64