Using the FTOC to find the derivative of the integral. I'm apologizing ahead of time because I don't know how to format an integral.
If I have the following integral: $$\int_{x^2}^5 (4x+2)\;dx.$$
I need to find the derivative, so could I do the following?
Multiply the integral by -1 and swap the limits of integration so that they are from 5 to x^2. 
Then, use FTOC and say that the derivative will be $-((4x^2)+2))$.
Is this correct?

You have
$$
f(x) = \int_{x^2}^5 4t + 2 \; dt
$$
and you want to find $f'(x)$.
First note that 
$$
f(x) = -\int_5^{g(x)} 4t + 2\; dt
$$
where $g(x) = x^2$. So $f(x)$ is the composition of two functions:
$$
f(x) = h(g(x)).
$$
where
$$\begin{align}
h(x) &= -\int_5^x 4x + 2\; dt \quad \text{and}\\
g(x) &= x^2.
\end{align}
$$
So by the chain rule you have
$$
f'(x) = h'(g(x))\color{red}{g'(x)} = -(4g(x) + 2)\color{red}{g'(x)} = \dots
$$
It looks like you forgot the derivative of the inner function.
