Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y

Theorem
The indeterminate Diophantine equation:
 * $16x^2 + 32x + 20 = y^2 + y$

has exactly $4$ solutions:
 * $\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$

Proof
$17$ is prime and therefore the solution of only two sets of integer products:

This leaves us with four systems of equations with four solutions:

Hence the solution:
 * $\tuple {0, 4}$

Hence the solution:
 * $\tuple {-2, 4}$

Hence the solution:
 * $\tuple {0, -5}$

Hence the solution:
 * $\tuple {-2, -5}$

Also see

 * Diophantine Equation