$$\textbf{Step 1: Subtract } {x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3} \textbf{ from } {x}^{3}+{y}^{3}$$
$$\begin{aligned}&x^{3}+y^{3}-\left(x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\right) \\&\quad=x^{3}+y^{3}-x^{3}-3 x^{2} y-3 x y^{2}-y^{3}\end{aligned}$$
$$\textbf{Step 2:Combine the like terms and simplify}$$
$$\begin{aligned}&=\left(x^{3}-x^{3}\right)+\left(y^{3}-y^{3}\right)-3 x^{2} y-3 x y^{2} \\&=>x^{3}+y^{3}-\left(x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\right)=-3 x^{2} y-3 x y^{2}\end{aligned}$$
$$\textbf{Hence, } -3 x^{2} y-3 x y^{2} \textbf{ should be added to } x^{3}+3 x^{2} y+3 x y^{2}+y^{3} \textbf{ to get } x^{3}+y^{3}$$