<s>[INST] Finding all values in $\mathbb R$ for quadratic absolute value equation The question:

Determine the solution set (in $\mathbb R$) for the equation $|x^2+2x+2| = |x^2-3x-4|$

So far, I have determined that for this to be true, $|x^2-3x-4|$ must be greater or equal to $0$, giving $(x-4)(x+1)\ge0$. To find the solution set, $|x^2+2x+2|\ge-1$ as indicated by the roots of the RHS equation, but this is where I get stuck. 
Where am I going wrong?
 [/INST] Since $|x| = x$ or $-x$, $$|x^2+2x+2| = |x^2-3x-4|$$ if and only if
$$x^2+2x+2 = x^2-3x-4$$ or 
$$-(x^2+2x+2) = x^2-3x-4$$ or
$$x^2+2x+2 = -(x^2-3x-4)$$ or
$$-(x^2+2x+2) = -(x^2-3x-4)$$
Some cases are equivalent (can you guess which and why?), so the computations are easier than they look.
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