Find all integral solutions of the equation $\displaystyle x^4+2x^3+2x^2+2x+5=y^2$.

My attempt:

The equation can be re-written as

$\displaystyle (x^2+x+1)^2-x^2=y^2-2^2$

$\displaystyle (x^2+2x+1)(x^2+1)=(y+2)(y-2)$

$\displaystyle (x+1)^2(x^2+1)=(y+2)(y-2)$

By inspection (-1, 2), (-1, -2), (2, 7) and (2, -7) are solutions. Are these the only solutions?