To make the trinomial a perfect square, we want the coefficient of the $x$ term to be half of the coefficient of the $x^2$ term. So, we want $\frac{15}{2}$ to be the coefficient of $x$. This means that $c$ must be equal to $\left(\frac{15}{2}\right)^2 = \frac{225}{4}$. Therefore, the answer is $\boxed{\textbf{(D) } 225-fourths}$.
