Find all harmonic polynomials of order 3 in two dimensions.
$p(x,y)= a+ bx+cy+dxy+ex^2y+fxy^2+gx^3+hy^3+i*x^2+j*y^2$
$ \frac{\partial p}{\partial x}= b+dy+2exy+fy^2+3gx^2+2*ix$
$ \frac{\partial^2 p}{\partial x \partial x}= 2ey+6gx+2*i$
$\frac{\partial^2 p}{\partial y \partial y}= 2fx+6hy+2*j$
$0=\frac{\partial^2 p}{\partial x \partial x}+\frac{\partial^2 p}{\partial y \partial y}= 2fx+6hy+2ey+6gx+ 2*i + 2*j \Rightarrow 0=fx+3hy+ey+3gx+i+j$
$\iff 0=i+j, 0=f+3g, 0=3h+e $
So i get: $p(x,y)= a+bx+cy+dxy-3hx^2y-3gxy^2+gx^3+hy^3+ix^2-iy^2)$
I think, i don´t solve the question of the exercise correctly. Can i give more precious polynomials? Or what I should do on this exercise?
Let us consider the question on a different viewpoint.
Let us write your expression as a sum of homogeneous polynomials with resp. degree $0,1,2,3: $
$$p(x)= a+ (bx+cy)+(dxy+\alpha x^2+\beta y^2)+(ex^2y+fxy^2+gx^3+hy^3)$$
(I have added the omitted terms as signaled by @Will Nagy). Let us consider the 3rd degree part.
You may know that harmonic functions can always be considered either as the real part, or as the imaginary part of a holomorphic function (due to Cauchy-Riemann equations). The 3rd degree polynomial functions are not that numerous: you have to turn to the real and imaginary parts of
$$\tag{1} z^3=k(x+iy)^3=k(x^3-3xy^2)+i k(3x^2y-y^3)$$
(where $k$ is any real constant). You recognize here the coefficients "-3" of your own computation.
To the real or the imaginary part of $(2)$, you must add (with the same "recipe") a degree 2 harmonic function, etc.