Answer
Elijah was wrong because he did not write the equation in $x^2+bx+c=0$ form before factoring.
The solution set is $\left\{-4, -1\right\}$.
Work Step by Step
Elijah was wrong because he did not write the equation in the form $x^2+bx+c=0$ before factoring. The Zero-Product Property can only be used for equations in the form $ab=0$.
The correct work is the one below:
\begin{align*}
x^2+5x+6-2&=0\\
x^2+5x+4&=0\\
(x+4)(x+1)&=0\\
x+4&=0 &\text{or}& &x+1=0\\
x&=-4 &\text{or}& &x=-1\end{align*}