Prove that $x=-1$ if and only if $x^3+x^2+x+1=0$

Question

Let $x\in\mathbb{R}$. Prove that $x=-1$ if and only if $x^3+x^2+x+1=0$. This is a bi-conditional statement, thus to prove it we need to prove:

Answer 1

$$x=-1\implies x^3+x^2+x+1=(-1)^3+(-1)^2+-1+1=-1+1-1+1=0.$$

Answer 2

You did the hard(er) part.

To prove the other implication, you simply need to show that $x=-1$ satisfies the cubic equation.

Substitute and simplify!