This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$.
Let $f: A_1^2\to A_1$ be the map defined by $f(a,b)=ad+bx$. Show that $f$ is a surjection and that its kernel is isomorphic to the left ideal $A_1d^2+A_1(xd-1)$.
I can get f is a surjection, but it seems that $d^2$ is not in the kernel.
This seems to be nothing more but calculations.
We have $\ker f=\{(a,b):ad+bx=0\}$ and define $\phi:\ker f\to A_1d^2+A_1(xd-1)$ by $\phi(a,b)=ad^2+b(xd-1)$. Note that $\phi$ is a homomorphism and let's check that $\phi$ is bijective.
If $\phi(a,b)=0$ then $ad^2+b(xd-1)=0$. Since $ad+bx=0$ we get $ad^2-ad^2-b=0$, so $b=0$, and then from $ad=0$ we have $a=0$.
If $\alpha d^2+\beta(xd-1)\in A_1d^2+A_1(xd-1)$, then we have to show that there are $a,b\in A_1$ such that $ad+bx=0$ and $\alpha d^2+\beta(xd-1)=ad^2+b(xd-1)$. From the last two equations I found $b=-\alpha d^2-\beta(xd-1)$ and $a=\alpha(dx+1)+\beta x^2d$.