We will first evaluate the inner most integral, $\int_0^{x+1} (4x + 7y) dy$. We are integrating with respect to $y$, and treating $x$ as a constant.
$\int_0^{x+1} (4x + 7y) dy = [4xy + \frac{7}{2}y^2]_0^{x+1}$
$= [4x(x + 1) + \frac{7}{2}(x + 1)^2] - [4x(0) + \frac{7}{2}(0)^2]$
$= \frac{15}{2}x^2 + \text{____}x + \frac{7}{2}$