Let $f(x) =(x + 1)(x^2 + 1)$ and $g(x)=x^3+1$. We have

$$f(x)=g(y); \; f(y)=g(x)$$We know that $g(x)=x^3+1$ is a strictly increasing function since

$$ g(x_1)>g(x_2) \Longleftrightarrow x_1^3-x_2^3>0$$