Find the values for $a,b,c,d$ 
Given  $$x^3+4x^2y+axy^2+3xy-bx^cy+7xy^2+dxy+y^2=x^3+y^2$$  for any
  real number $x$ and $y$ , find the value of $a,b,c,d$


$$x^3+4x^2y+axy^2+3xy-bx^cy+7xy^2+dxy+y^2=x^3+y^2$$
R.H.S only has $x^3$ and $y^2$, therefore all the terms in L.H.S must cancel out except for $x^3$ and $y^2$.
So let's pair the terms that may cancel out.
$$x^3+y^2+(axy^2+7xy^2)+(3xy+dxy)+4x^2y-bx^cy$$
From here you can conclude that,
$a=-7,d=-3$
Also for $4x^2y$ to cancel $c=2$ and $b=4.$
