sat-dog

Choice C is correct. For the given equation, $\left(x+5\right)$ is a factor of both terms on the left-hand side. Therefore, the given equation can be rewritten as $\left(\frac{1}{4}-\frac{1}{3}\right)\left(x+5\right)=-7$, or $\left(\frac{3}{12}-\frac{4}{12}\right)\left(x+5\right)=-7$, which is equivalent to $-\frac{1}{12}\left(x+5\right)=-7$. Multiplying both sides of this equation by $-12$ yields $x+5=84$. Subtracting $5$ from both sides of this equation yields $x=79$. 

Choice A is incorrect. This is the value of $x$ for which the left-hand side of the given equation equals $\frac{7}{12}$, not $-7$.

Choice B is incorrect. This is the value of $x$ for which the left-hand side of the given equation equals $0$, not $-7$.

Choice D is incorrect. This is the value of $x$ for which the left-hand side of the given equation equals $-\frac{209}{12}$, not $-7$.