**SOLUTION**

Wehave

$$

\begin{aligned}

& a^{3}-b^{3}+1+3 a b \\

=& a^{3}+(-b)^{3}+(1)^{3}-3 \times a \times(-b) \times 1 \\

=& x^{3}+y^{3}+z^{3}-3 x y z, \text { where } a=x,(-b)=y \text { and } 1=z \\

=&(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right) \\

=&(a-b+1)\left(a^{2}+b^{2}+1+a b+b-a\right) \\

=&(a-b+1)\left(a^{2}+b^{2}+a b-a+b+1\right) .

\end{aligned}

$$