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