Why can't you solve $0 = -5\sqrt{x+2}-2$ So i did this,
\begin{aligned}
2&amp;=-5\sqrt{x+2},\\
\tfrac{2}{-5}&amp;=\sqrt{x+2},\\
\tfrac{4}{25}&amp;=x+2,\\
x&amp;=-\tfrac{46}{25}.
\end{aligned}
Is there something wrong with my algebra?
Remember that, $\sqrt{x+2}≥0$ and multiplying each side by $(-1)$  then you get,
$$-5\sqrt{x+2}-2=0 \Longleftrightarrow 5\sqrt{x+2}+2=0$$ which is impossible.
The sum of  non-negative and positive  numbers can not equal to zero.
EDIT:
Your argument is not generally correct. Because, $$a=b \not \Longleftrightarrow a^2=b^2.$$ For example,
$$-1=1 \\  (-1)^2=1^2 \\  1=1 $$ which is invalid.
You've incorrectly assumed $f(x)=f(y)\implies x = y$ (line 2 to line 3) which is not true when $f$ is non injective.