By the [Triangle Inequality](https://artofproblemsolving.com/wiki/index.php/Triangle_Inequality), $PR + QR \ge PR$, and equality holds if $R$ is on $PQ$.  The equation of the line with $P$ and $Q$ is $y = \frac{4}{5}x - \frac{6}{5}$, so point $R$ is $(1,-\frac{2}{5})$.  Thus, $m = \boxed{\textbf{(B) } -\frac{2}{5}}$.
Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals:
$\text{(A) } -\tfrac{3}{5}\quad \text{(B) } -\tfrac{2}{5}\quad \text{(C) } -\tfrac{1}{5}\quad \text{(D) } \tfrac{1}{5}\quad \text{(E) either }-\tfrac{1}{5}\text{ or} \tfrac{1}{5}.$