4. Determine the largest value of $x$ for which
$$
\left|x^{2}-11 x+24\right|+\left|2 x^{2}+6 x-56\right|=\left|x^{2}+17 x-80\right| \text {. }
$$
4. Answer: 8
Since $2 x^{2}+6 x-56-\left(x^{2}-11 x+24\right)=x^{2}+17 x-80$, the given equation holds if and only if
$$
\left(x^{2}-11 x+24\right)\left(2 x^{2}+6 x-56\right) \leq 0 \text {, }
$$

Since $|a-b|=|a|+|b|$ if and only if $a b \leq 0$. The above inequality reduces to
$$
(x-3)(x-8)(x-4)(x+7) \leq 0 \text {. }
$$

Since
$$
\{x:(x-3)(x-8)(x-4)(x+7) \leq 0\}=[-7,3] \cup[4,8]
$$
we conclude that the largest value of $x$ is 8 .