Try this beautiful problem from TOMATO Objective no. 257 based on Roots of a Quintic Polynomial.
Problem: Roots of a Quintic Polynomial
The number of real roots of [latex] x^5+2x^3+x^2+2=0[/latex] is
(A) 0
(B) 3
(C) 5
(D) 1
Solution: Answer: (D)
[latex] x^5+2x^3+x^2+2=0[/latex]
[latex] \implies x^3(x^2+2)+(x^2+2)=0[/latex]
[latex] \implies (x^3+1)(x^2+2)=0[/latex]
[latex] \implies (x+1)\bold{\underline{(x^2-x+1)(x^2+2)}}=0[/latex]
The expression in underline doesn't have any real roots.
Therefore, only real root of the equation is [latex] x=-1[/latex]
