Solution - Reducing fractions to their lowest terms

Step  1  :

            30
 Simplify   ——
            x 

Equation at the end of step  1  :

         30     
  (6x -  ——) -  5
         x      

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

           6x     6x • x
     6x =  ——  =  ——————
           1        x   

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 :

 2.2       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:

 6x • x - (30)     6x2 - 30
 —————————————  =  ————————
       x              x    

Equation at the end of step  2  :

  (6x2 - 30)    
  —————————— -  5
      x         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

         5     5 • x
    5 =  —  =  —————
         1       x  

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   6x² - 30  =   6 • (x² - 5) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x² - 5 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 5 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 6 • (x2-5) - (5 • x)     6x2 - 5x - 30
 ————————————————————  =  —————————————
          x                     x      

Trying to factor by splitting the middle term

 4.4     Factoring  6x² - 5x - 30 

The first term is,  6x²  its coefficient is  6 .
The middle term is,  -5x  its coefficient is  -5 .
The last term, "the constant", is  -30 

Step-1 : Multiply the coefficient of the first term by the constant   6 • -30 = -180 

Step-2 : Find two factors of  -180  whose sum equals the coefficient of the middle term, which is   -5 .

For tidiness, printing of 12 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 :

  6x2 - 5x - 30
  —————————————
        x