$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= 1 + y + x^2 + x^2\,y \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= 1 + x^2 + \left( 1 + x^2 \right) \,y \\ \frac{\mathrm{d}y}{\mathrm{d}x} - \left( 1 + x^2 \right) \, y &= 1 + x^2 \end{align*}$
Now since this is first order linear you can solve with an integrating factor...