Here is the question:
To estimate the proportion $p_1$ of male voters who are in favor of expanding the use of solar energy, take a random sample of size $m$ and set $X$ for the number in favor. To estimate the corresponding proportion $p_2$ of female voters, take an independent random sample of size $n$ and set $Y$ for the number in favor.
Set $\hat{p_1} = X/m$ and $\hat{p_2} = Y/n$. Give the standard error of $\hat{p_1} - \hat{p_2}$.
I modeled X and Y as binomial distributions; here is my work:
$$
\begin{align}
\sigma_{\hat{p_1} - \hat{p_2}} &= \sqrt{Var(\hat{p_1} - \hat{p_2})} \\
&= \sqrt{Var(\hat{p_1}) + Var(\hat{p_2})} \\
&= \sqrt{Var(\frac{X}{m}) + Var(\frac{Y}{n})}\\
&= \sqrt{\frac{1}{m} Var(X) + \frac{1}{n} Var(Y)}\\
&= \sqrt{\frac{1}{m} (mp_1(1-p_1)) + \frac{1}{n}(np_2(1-p_2))} \\
&= \boxed{\sqrt{p_1(1-p_1) + p_2(1-p_2)}}
\end{align}
$$
However, the given solution is different:
Why is there an extra $m$ and extra $n$ in the solution?
Ah, I see my problem.
$$ Var(aX) = a^2Var(X)$$.
I was forgetting to take that into account when taking $\frac{1}{m}$ and $\frac{1}{n}$ out of the variance.