jmc

Using the quadratic formula, we find that the roots of the quadratic are $\frac{-7\pm \sqrt{7^2-4(4)(k)}}{8}=\frac{-7\pm \sqrt{49-16k}}{8}$. Since the problem tells us that these roots must equal $\frac{-7\pm \sqrt{15}i}{8}$, $$\begin{array}{rl}\sqrt{49-16k}& =\sqrt{15}i\\ \Rightarrow \sqrt{49-16k}& =\sqrt{-15}\\ \Rightarrow 49-16k& =-15\\ \Rightarrow 16k& =64\\ \Rightarrow k& =\boxed{4}.\end{array}$$