Solution - Simplification or other simple results

Step  1  :

            d
 Simplify   —
            d

Equation at the end of step  1  :

  (1 • x) • (x3 + x2 + x + 1)

Step  2  :

Checking for a perfect cube :

 2.1    x³+x²+x+1  is not a perfect cube

Trying to factor by pulling out :

 2.2      Factoring:  x³+x²+x+1 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  x+1 
Group 2:  x³+x² 

Pull out from each group separately :

Group 1:   (x+1) • (1)
Group 2:   (x+1) • (x²)
               -------------------
Add up the two groups :
               (x+1)  •  (x²+1) 
Which is the desired factorization

Polynomial Roots Calculator :

 2.3    Find roots (zeroes) of :       F(x) = x²+1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  x • (x2 + 1) • (x + 1)