2nd order nonhomogenous ODE with delta function (Green's Fn)

Question

I am trying to solve $\frac{u'}{(1+x^2)^2}-\frac{u''}{1+x^2}=\delta(x-\alpha), u(0)=u(1)=0$

Wolfram alpha gives $u(x)=-1/3(x^3+3x)(\sigma(x)-1)$

I have no idea how to set up an ansatz. Maybe some property of green's function would help.

Many thanks.