Here is a link

Exact differential equation - Wikipedia, the free encyclopedia
This should also be in your book.

An equation is exact if there is a function $\displaystyle f(x,y)$ such that

$\displaystyle \nabla f \cdot d\vec{r}=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}=0$

Where $\displaystyle \vec{r}=x\vec{i}+y\vec{j}$ is the position vector.

Since the mixed partials match we know such a function exists so

$\displaystyle \frac{\partial f}{\partial y}=(x+y+1)^2 \implies f(x,y)=\frac{1}{3}(x+y+1)^3+g(x)$

But we also know what the partial derivative with respect to $\displaystyle x$ must be so we can solve for $\displaystyle g(x)$

$\displaystyle \frac{\partial f}{\partial x}=(x+y+1)^2+g'(x)=(x+y+1)^2+x^3 \implies g(x)=\frac{1}{4}x^4$

This gives $\displaystyle \frac{1}{3}(x+y+1)^3+\frac{1}{4}x^4=c$

is an implicit solution to the equation.