collect a multivariate polynomial, and simplify all coefficients

I have a polynomial like this:

(a x + b c y + d z) (d x + e c y + g z)

I would like to rewrite it as something like this:

a d x^2 + b c^2 e y^2 + d g z^2 + (c (b d + a e)) x y + (d^2 + a g) x z + (c (d e + b g)) y z

i.e. keep all terms of different variables separate, collect all the coefficients for each term, and also simplify all the coefficients. How can we do this?

Collect gives a nested collection, so it is not what I want.

(i.e. it gives this:

a d x^2 + b c^2 e y^2 + (c d e + b c g) y z + d g z^2 + x ((b c d + a c e) y + (d^2 + a g) z)

where the last term is a nested one)

The closest I can find is

expr=(a x + b c y + d z) (d x + e c y + g z);

Plus @@ MonomialList[Expand[expr], {x, y, z}]

which collects all the coefficients but refuses to simplify them, i.e.

a d x^2 + (b c d + a c e) x y + b c^2 e y^2 + (d^2 + a g) x z + (c d e + b c g) y z + d g z^2.