Prove or disprove following equation has infinitely many integer solutions in $\mathbb Z$:
$x^2+y^2-29t^2-40t=29$
$x^2+y^2=29t+40t+29=(25t^2+20t+4)+(4t^2+20t+25)$
$x^2=(5t+2)^2$ ⇒ $x=±(5t+2)$
$y^2=(2t+5)^2$ ⇒ $y=±(2t+5)$
The equation is symmetric for x and y, so are the solutions.Hence this equation has infinitely many solutions.
Is this solution reasonable? Is there a direct method for prove or disprove this?