If -5 is a root of the quadratic equation 2x^2 + px - 15 = 0 and the quadratic equation p(x^2 + x) + k = 0 has equal roots, find the value of k.

Since $-5$ is a root of $2x^2 + px - 15 = 0$, substitute $x = -5$: $2(-5)^2 + p(-5) - 15 = 0 \implies 50 - 5p - 15 = 0 \implies 35 = 5p \implies p = 7$. Substituting $p = 7$ in the second equation gives $7(x^2 + x) + k = 0 \implies 7x^2 + 7x + k = 0$. For equal roots, discriminant $D = 0 \implies 7^2 - 4(7)(k) = 0 \implies 49 - 28k = 0 \implies k = \frac{49}{28} = \frac{7}{4}$.