Not of Bernoulli type, but still linear.
[tex](1+x^2)\dfrac{\mathrm dy}{\mathrm dx}+2xy=4x^2[/tex]
There's no need to find an integrating factor, since the left hand side already represents a derivative:
[tex]\dfrac{\mathrm d}{\mathrm dx}[(1+x^2)y]=(1+x^2)\dfrac{\mathrm dy}{\mathrm dx}+2xy[/tex]
So, you have
[tex]\dfrac{\mathrm d}{\mathrm dx}[(1+x^2)y]=4x^2[/tex]
and integrating both sides with respect to [tex]x[/tex] yields
[tex](1+x^2)y=\displaystyle\int4x^2\,\mathrm dx[/tex]
[tex](1+x^2)y=\dfrac43x^3+C[/tex]
[tex]y=\dfrac{4x^3}{3(1+x^2)}+\dfrac C{1+x^2}[/tex]
