OME 1992 Problem 3

Prove that if $a,b,c,d$ are nonnegative integers satisfying $(a+b)^2+2a+b= (c+d)^2+2c+d$ , then $a = c$ and $b = d$ . Show that the same is true if $a,b,c,d$ satisfy $(a+b)^2+3a+b=(c+d)^2+3c+d$ , but show that there exist $a,b,c,d$ with $a \ne c$ and $b \ne d$ satisfying $(a+b)^2+4a+b = (c+d)^2+4c+d$ .