<s>[INST] 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 $a,b,c,d$

 [/INST] Put some values of $x$ and $y$ in it.
Say $x=1$:  $$(a+7)y^2+(7-b+d)y =0\;\;\forall y$$ so $a=-7$ and $b-d=7$. 
Now we have (for all $x,y$):
$$4x^2y-bx^cy+(d+3)xy=0$$
If $y\ne 0$ we get $$4x^2-bx^c+(d+3)x=0\;\;\;\forall x$$

a) Put $x= 2$: $$\boxed{8+b(2-2^c)=0}$$
b) Put $x=4$: $$\boxed{48+b(4-4^c)=0}$$
Solving this system we get $$-b = {8\over 2-2^c} = {48\over 4-4^c}\implies 2+2^c= 6\implies c = 2$$ so $b=4$ and $d=-3$.
</s>