<s>[INST] Solve for $x$ in $\frac{x^2+12x+36}{x^2+4x-12}=2$ $\frac{x^2+12x+36}{x^2+4x-12}=2$
I factored top and bottom to $\frac{(x+6)(x+6)}{(x+6)(x-2)}=2$
and eliminated the common factor (x+6) for $\frac{x+6}{x-2}=2$
then $x+6=2(x-2)$  and  $x+6=2x-4$  and  $6+4=2x-x$ and finally $x = 10$
Plugging this back into the original fraction proves true
My instructor however in her exam prep test says
$\frac{x^2+12x+36}{x^2+4x-12}=2$
$$x^2+12x+36 = 2(x^2+4x-12)$$
$$x^2+12x+36 = 2x^2+8x-24$$
$$0 =2x^2-x^2 +8x -12x -24 -36$$
$$0 =x^2-4x -60$$ 
which she then factors to $(x-10)(x+6)$ saying the answer to the problem is $x=10$ or $x=-6$
But if you plug -6 back into the original fraction you get $0=2$. Where is the problem here? Thanks
 [/INST] In short, you are correct.
Slightly longer, the expression
$$\frac{x^2+12x+36}{x^2+4x-12}$$
is not defined for $x = {-6}$ because $x^2 + 4x -12 = 0$ when $x = -6$ and division by zero is undefined. Hence a priori to even talk about this fraction we assume that $x\neq -6$ and $x\neq2$.
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