Solution - Polynomial long division

Step  1  :

            x2 + x + 1
 Simplify   ——————————
                x     

Trying to factor by splitting the middle term

 1.1     Factoring  x² + x + 1 

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

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   1 .

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

Polynomial Long Division :

 1.2    Polynomial Long Division
Dividing :  x²+x+1 
                              ("Dividend")
By         :    x    ("Divisor")

Quotient :  x+1 
Remainder :  1 

Equation at the end of step  1  :

  (x2 + x + 1)    
  ———————————— +  1
       x          

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

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

         1     1 • x
    1 =  —  =  —————
         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:

 (x2+x+1) + x     x2 + 2x + 1
 ————————————  =  ———————————
      x                x     

Trying to factor by splitting the middle term

 2.3     Factoring  x² + 2x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +2x  its coefficient is  2 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   2 .

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

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 :
                     1 • (x+1)
Step-5 : Add up the four terms of step 4 :
                    (x+1)  •  (x+1)
             Which is the desired factorization

Multiplying Exponential Expressions :

 2.4    Multiply  (x+1)  by  (x+1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x+1)  and the exponents are :
          1 , as  (x+1)  is the same number as  (x+1)¹ 
 and   1 , as  (x+1)  is the same number as  (x+1)¹ 
The product is therefore,  (x+1)⁽¹⁺¹⁾ = (x+1)² 

Final result :

  (x + 1)2
  ————————
     x