The general solution of the differential equation

$\frac{dy}{dx} = xy + x + y + 1$ is

The given differential equation is:

$\frac{dy}{dx} = xy + x + y + 1$

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Factoring the right side by grouping terms:

$\frac{dy}{dx} = (xy + x) + (y + 1)$

$\frac{dy}{dx} = x(y + 1) + 1(y + 1)$

$\frac{dy}{dx} = (y + 1)(x + 1)$

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Separating the variables:

$\frac{dy}{y + 1} = (x + 1) dx$

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Integrating both sides:

$\int \frac{dy}{y + 1} = \int (x + 1) dx$

For the left side:

$\int \frac{dy}{y + 1} = \log_e|y + 1|$

For the right side:

$\int (x + 1) dx = \frac{x^2}{2} + x$

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The general solution is:

$\log_e|y + 1| = \frac{1}{2}x^2 + x + C$

where $C$ is the constant of integration.