Solution - Reducing fractions to their lowest terms

Step  1  :

            12
 Simplify   ——
            7 

Equation at the end of step  1  :

           12                    
  ((4x +  (—— • x2)) -  21x) -  126
           7                     

Step  2  :

Equation at the end of step  2  :

          12x2             
  ((4x +  ————) -  21x) -  126
           7               

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  7  as the denominator :

           4x     4x • 7
     4x =  ——  =  ——————
           1        7   

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:

 4x • 7 + 12x2     12x2 + 28x
 —————————————  =  ——————————
       7               7     

Equation at the end of step  3  :

   (12x2 + 28x)            
  (———————————— -  21x) -  126
        7                  

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  7  as the denominator :

           21x     21x • 7
    21x =  ———  =  ———————
            1         7   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   12x² + 28x  =   4x • (3x + 7) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 4x • (3x+7) - (21x • 7)     12x2 - 119x
 ———————————————————————  =  ———————————
            7                     7     

Equation at the end of step  5  :

  (12x2 - 119x)    
  ————————————— -  126
        7          

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  7  as the denominator :

           126     126 • 7
    126 =  ———  =  ———————
            1         7   

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   12x² - 119x  =   x • (12x - 119) 

Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions

 x • (12x-119) - (126 • 7)     12x2 - 119x - 882
 —————————————————————————  =  —————————————————
             7                         7        

Trying to factor by splitting the middle term

 7.3     Factoring  12x² - 119x - 882 

The first term is,  12x²  its coefficient is  12 .
The middle term is,  -119x  its coefficient is  -119 .
The last term, "the constant", is  -882 

Step-1 : Multiply the coefficient of the first term by the constant   12 • -882 = -10584 

Step-2 : Find two factors of  -10584  whose sum equals the coefficient of the middle term, which is   -119 .

For tidiness, printing of 42 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 :

  12x2 - 119x - 882
  —————————————————
          7