<s>[INST] How do you find the values that $x$ can take by squaring? $$|x+4| \cdot |x-4| = x+4$$
How do you find the values that $x$ can take by squaring? 
How I've tried
$$|x+4|^2 \cdot |x-4|^2 = (x+4)^2$$
and
$$|x^2+16| \cdot |x^2+16| = x^2+16$$
I think I went wrong. 
 [/INST] Given
$$
|x+4||x-4|=x+4,
$$
and we want to find $x$, we observe that 
$$
|x+4||x-4|=|(x+4)(x-4)|=|x^2+4x-4x-16|=|x^2-16|.
$$
There are two cases:


*

*$x^2-16\geq 0$.  This happens when $x\geq 4$ or $x\leq -4$.  When $x^2-16\geq 0$, $|x^2-16|=x^2-16$ since both are nonnegative.  Therefore, you want to solve $x^2-16=x+4$.  Now, use the quadratic formula, you can get the solutions (but you'll need to check them against the inequalities above).

*$x^2-16<0$.  This happens when $-4<x<4$.  When $x^2-16<0$, then $|x^2-16|=-(x^2-16)=16-x^2$.  This is true since $x^2-16<0$ but the absolute value makes it positive by multiplying by $-1$.  Again, you need to use the quadratic formula to solve $16-x^2=x+4$ (and again, discard any values that you get that violate $-4<x<4$).
By other's observations on the positivity, you only need to consider the case where $x\geq -4$, but that only slightly simplifies this approach.
</s>