How to simplify this expression using tensor notaion?

Question

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$

Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.

Answer 1

$\nabla^2 (\phi \vec A)=\partial_i^2(\phi A_j)=\phi \partial_i^2(A_j)+2\partial_i (\phi) \partial_i(A_j)+A_j\partial_i^2(\phi)=\phi \nabla^2 \vec A+2(\nabla \phi \cdot \nabla)\vec A+\vec A \nabla^2 \phi$

And the rest is trivial.