Solution:
The only solutions are $x, y = -1, 1; 0, -1; 0, 0; -6, 2; 5, 2$.

$(y^{2} + y / 2 - 1 / 2)(y^{2} + y / 2 + 1 / 2) = y^{4} + y^{3} + \frac{1}{4}y^{2} - \frac{1}{4} < y^{4} + y^{3} + y^{2} + y$ except for $-1 \leq y \leq -1/3$.

Also $(y^{2} + y / 2)(y^{2} + y / 2 + 1) = y^{4} + y^{3} + \frac{5}{4}y^{2} + y / 2$ which is greater than $y^{4} + y^{3} + y^{2} + y$ unless $0 \leq y \leq 2$.

But no integers are greater than $y^{2} + y / 2 - 1 / 2$ and less than $y^{2} + y / 2$. So the only possible solutions have $y$ in the range $-1$ to $2$. Checking these 4 cases, we find the solutions listed.