<s>[INST] Trying to solve $|2x-15| = -x^2 - 5x -8$ My first instinct was to take the positive and negative of the right hand side, resulting in
$2x-15 = -x^2 - 5x - 8$, and $2x-15 = x^2 + 5x + 8$, which results in the first giving me two real answers using the quadratic equation, and the second being two imaginary solutions. The problem is that, when graphed, these 2 graphs do not intersect at all, so there should be no real solutions. As for the given complex solutions, neither were considered by wolfram alpha.
Knowing that there are no real solutions, I'm confused as to how I might get complex solutions through means not already attempted and explained above.
 [/INST] I would recommend you to notice that
\begin{align*}
x^{2} + 5x + 8 = \left(x^{2} + 5x + \frac{25}{4}\right) + \frac{7}{4} = \left(x + \frac{5}{2}\right)^{2} + \frac{7}{4} \geq \frac{7}{4}
\end{align*}
Hence we conclude that $-x^{2} - 5x - 8 < 0$ for every possible value of $x$.
Based on such considerations, we conclude the proposed equation has no solutions in $\mathbb{R}$.
Hopefully this helps!
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