Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            4
 Simplify   —
            5

Equation at the end of step  1  :

      (x2)       x   4
  ((0-————)+(31•——))+—
       40       40   5

Step  2  :

             x
 Simplify   ——
            40

Equation at the end of step  2  :

      (x2)       x   4
  ((0-————)+(31•——))+—
       40       40   5
 Step  3  :
            x2
 Simplify   ——
            40

Equation at the end of step  3  :

         x2     31x     4
  ((0 -  ——) +  ———) +  —
         40     40      5

Step  4  :

Adding fractions which have a common denominator :

 4.1       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 -x2 + 31x     31x - x2
 —————————  =  ————————
    40            40   

Equation at the end of step  4  :

  (31x - x2)    4
  —————————— +  —
      40        5

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   31x - x²  =   -x • (x - 31) 

Calculating the Least Common Multiple :

 6.2    Find the Least Common Multiple

      The left denominator is :       40 

      The right denominator is :       5 

      Least Common Multiple:
      40 

Calculating Multipliers :

 6.3    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 1

   Right_M = L.C.M / R_Deno = 8

Making Equivalent Fractions :

 6.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      -x • (x-31)
   ——————————————————  =   ———————————
         L.C.M                 40     

   R. Mult. • R. Num.      4 • 8
   ——————————————————  =   —————
         L.C.M              40  

Adding fractions that have a common denominator :

 6.5       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:

 -x • (x-31) + 4 • 8     -x2 + 31x + 32
 ———————————————————  =  ——————————————
         40                    40      

Trying to factor by splitting the middle term

 6.6     Factoring  -x² + 31x + 32 

The first term is,  -x²  its coefficient is  -1 .
The middle term is,  +31x  its coefficient is  31 .
The last term, "the constant", is  +32 

Step-1 : Multiply the coefficient of the first term by the constant   -1 • 32 = -32 

Step-2 : Find two factors of  -32  whose sum equals the coefficient of the middle term, which is   31 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  32 
                     -x² - 1x + 32x + 32

Step-4 : Add up the first 2 terms, pulling out like factors :
                    -x • (x+1)
              Add up the last 2 terms, pulling out common factors :
                    32 • (x+1)
Step-5 : Add up the four terms of step 4 :
                    (-x+32)  •  (x+1)
             Which is the desired factorization

Final result :

  (x + 1) • (32 + x)
  ——————————————————
          40