<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] You can split the problem into 3 cases: $x\lt-4$, $-4\le x\le4$, and $4\lt x$.
For $x \lt -4$:


*

*There are no solutions for this case because $|x+4|*|x-4|$ is always positive, and $x+4$ will always be negative for $x<-4$.


For $-4\le x\le 4$:


*

*$|x-4| = 4-x$, and $|x+4| = x+4$, so solve $(4-x)*(x+4)=x+4$.

*$-x^2+16 = x+4$

*$0=x^2+x-12$

*$x = 3, -4$
For $4\lt x$:


*

*$|x-4| = x-4$, and $|x+4| = x+4$, so solve $(x-4)*(x+4)=x+4$.

*$x^2-16 = x+4$

*$0=x^2-x-20$

*$x = 5, -4$
All solutions: $x = -4, 3, 5$
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