Setting $x = -2$ in the equation $Q_1(x) (x + 2) - 13 = Q_3(x) (x + 2)(x^2 - 3x - 4) + R(x),$ we get
\[R(-2) = -13.\]Setting $x = 4$ and $x = -1$ in the equation $Q_2(x) (x^2 - 3x - 4) - 5x - 11 = Q_3(x) (x + 2)(x^2 - 3x - 4) + R(x),$ we get
\[R(4) = -31 \quad \text{and} \quad R(-1) = -6.\]Since $\deg R(x) = 2,$ we can let $R(x) = ax^2 + bx + c.$  Then
\begin{align*}
4a - 2b + c &= -13, \\
16a + 4b + c &= -31, \\
a - b + c &= -6.
\end{align*}Subtracting these equations in pairs, we get
\begin{align*}
12a + 6b &= -18, \\
3a - b &= -7.
\end{align*}Solving, we find $a = -2$ and $b = 1,$ so $c = -3.$  Hence, $R(x) = -2x^2 + x - 3.$
Let $P(x),$ $Q_1(x),$ $Q_2(x),$ $Q_3(x),$ $R(x)$ be polynomials such that
\begin{align*}
P(x) &= Q_1(x) (x + 2) - 13 \\
&= Q_2(x) (x^2 - 3x - 4) - 5x - 11 \\
&= Q_3(x) (x + 2) (x^2 - 3x - 4) + R(x),
\end{align*}and $\deg R(x) = 2.$  Find $R(x).$