The question I am struggling with is asking how many unique solutions there are to the following:
$$(x-2)^2 (x+2)^2 = 4+\log(x+4)$$
Wolfram Alpha tells me the answer is four, but as far as I can figure out, you could not solve for these four solutions without a graphical calculator or from what I've gathered from Google, using the Lambert W function. But, I do not need to solve it - I just need to prove/state how many unique solutions there are.
Expanding/simplifying etc gets me to:
$$12 e^x+x = e^{12}-4$$
By looking at this, is there a way to prove this has four unique solutions? This is an A-level equivalent question so I assume the answer is not too complex, however I am stumped!
Hint: We rewrite the problem as finding the zeros of $(x-2)^2(x+2)^2-(4+log(x+4))$. Looking at the graph below, specifically the derivative at different regions, could you conclude that there won't be a fifth zero?