Consider the quadratic equation $x^2+x+1=0$.

$x=0$ is not a solution to this equation, so it is safe to divide both sides by $x$.

$x+1+\frac{1}{x}=0$

Rearrange:

$x+1=-\frac{1}{x}$

And substitute into the original equation.

$x^2-\frac{1}{x}=0$

Multiply both sides by $x$.

$x^3-1=0$

Trivially, a solution of this equation is $x=1$ (as $1^3-1=0$).

Substitute into the original equation:

$1^2+1+1=0$

$\therefore3=0$

What's gone wrong here?

Attribution to be posted after the question is solved.

Consider the quadratic equation $x^2+x+1=0$.

$x=0$ is not a solution to this equation, so it is safe to divide both sides by $x$.

$x+1+\frac{1}{x}=0$

Rearrange:

$x+1=-\frac{1}{x}$

And substitute into the original equation.

$x^2-\frac{1}{x}=0$

Multiply both sides by $x$.

$x^3-1=0$

Trivially, a solution of this equation is $x=1$ (as $1^3-1=0$).

Substitute into the original equation:

$1^2+1+1=0$

$\therefore3=0$

What's gone wrong here?

Edit for attribution: I found the puzzle on the Mind Your Decisions blog.