$$\begin{align} f&:=\text{two variable function of }(x,y)\\ \Delta f&:= {\partial^2 f \over \partial x^2}+{\partial^2 f \over \partial y^2}\\ i&:=\text{imaginary number}\\ f(x,y)&:=\Re\left\{ (x^2-iy^2)(1+i)+ \exp(x+iy ) \right\}\\ &= \underbrace{\Re\left\{ x^2+x^2 i-iy^2-i^2 y^2+ \exp(x+iy ) \right\}}_{\text{My works begins from this line} } \\ &=\Re\left\{ x^2+x^2 i-iy^2+ y^2+ \exp(x+iy ) \right\}\\ &=\Re\left\{ x^2+ y^2+ \underbrace{ \color{red}{\exp(x+iy )} }_{\text{How can I analyze this?} } \right\}\\ \end{align}$$
The final objective for me is to evaluate $\Delta f$ from the given equation.
$e^{x+iy}=e^x\,e^{iy}$
$e^{iy}=(\cos y+ i\sin y)$
Hence
$e^{x+iy}=(\cos y+ i\sin y)e^x$
Can you get it from here?