Consider a function $f(x, y) = Ax^5 + Bx^4y + Cx^3y^2 + Dx^2y^3 + xy^4 − y^5$ 
Consider a function $f(x, y) = Ax^5 + Bx^4y + Cx^3y^2 + Dx^2y^3 + xy^4 − y^5$
where $A$, $B$, $C$, $D$ are unspecified real numbers. Determine the values of
$A$, $B$, $C$, $D$ such that $f(x,y)$ satisfies $f_{xx}(x,y) + f_{yy}(x,y) = 0$

What I did so far, I took double derivative of x and y:

$f_{xx}(x,y) = 20Ax^3+12Bx^2y+6Cxy^2+2Dy^3$
$f_{yy}(x,y) = 2Cx^3+6Dx^2y+12xy^2-20y^3$

How can I satisfies $f_{xx}(x,y) + f_{yy}(x,y) = 0$? There are no terms that cancels, subtracts or add.

You've calculated $f_{xx},f_{yy}$. Now, note that $f_{xx}+f_{yy}$ is a polynomial, so $f_{xx}+f_{yy}=0$ for all $x,y$ if and only if all the coefficients are zero. This will give you a set of conditions on $A,B,C,D$.
