$(a).$
$$\begin{align}\frac{\partial u(x,y)}{\partial x}&=3x^2-3y^2+3\\ \frac{\partial^2u(x,y)}{\partial x^2}&=6x\ \\ \frac{\partial u(x,y)}{\partial y}&=-6xy+2\\ \frac{\partial^2u(x,y)}{\partial y^2}&=-6x\end{align}$$
Where first and second partial derivatives are continuous with respect to both $x$ and $y$.
$$\Delta u= \frac{\partial^2u(x,y)}{\partial x^2}+ \frac{\partial^2u(x,y)}{\partial y^2}= 6x+(-6x)=0$$
Therefore, $u(x,y)=x^3-3xy^2+2y+3x$ is a harmonic function.
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$(b).$
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Use CR-equation to find the harmonic conjugate.
$$\frac{\partial u}{\partial x }=3x^2-3y^2+3= \frac{\partial v}{\partial y }$$
$$\begin{align}\Rightarrow v(x,y)&= \int(3x^2-3y^2+3)dy \\&=3x^2y-y^3+3y+h(x) \\ \Rightarrow \frac{\partial v}{\partial x }=6x+h'(x)\end{align}$$
Hence,$$\frac{\partial u}{\partial y } =-6xy+2=-\frac{\partial v}{\partial x }=-(6x+h'(x)) \\ \Rightarrow h'(x)=-2 \\ \Rightarrow h(x)= -2x$$
Therefore the harmonic conjugate $v(x,y)=3x^2y-y^3+3y-2x.$
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$(c).$
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$$\begin{align}u(x,y)+iv(x,y)&=x^3-3xy^2+2y+3x+i(3x^2y-y^3+3y-2x)\\ &=x^3-3xy^2+i3x^2y-iy^3+3x+i3y+2y-i2x\end{align}$$
Consider $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$
Thus $$(x+iy)^3=x^3+i3x^2y-3xy^2-iy^3=z^3$$
$$3x+i3y=3(x+iy)=3z$$
$$2y-i2x=-i2(x+iy)=-2iz$$
Therefore, $f(z)=z^3+3z-2iz$.
