Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (((g4)+(5•(g2)))+5)•(((5•(g4))-(2•5g2))+11g)
 Step  2  :

Equation at the end of step  2  :

  (((g4)+(5•(g2)))+5)•((5g4-(2•5g2))+11g)
 Step  3  :

Equation at the end of step  3  :

  (((g4) +  5g2) +  5) • (5g4 - 10g2 + 11g)

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   5g⁴ - 10g² + 11g  =   g • (5g³ - 10g + 11) 

Trying to factor by splitting the middle term

 5.2     Factoring  g⁴ + 5g² + 5 

The first term is,  g⁴  its coefficient is  1 .
The middle term is,  +5g²  its coefficient is  5 .
The last term, "the constant", is  +5 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 5 = 5 

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

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

Polynomial Roots Calculator :

 5.3    Find roots (zeroes) of :       F(g) = 5g³-10g+11
Polynomial Roots Calculator is a set of methods aimed at finding values of  g  for which   F(g)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  g  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  5  and the Trailing Constant is  11.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,11

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  g • (g4 + 5g2 + 5) • (5g3 - 10g + 11)