Prove that the ideal $M$ = $(x+2)F[x] + (x^2+8x+16)F[x]$ gives $F[x]$

Question

I claim that it suffices to show $M$ contains $f\in{F[x]}$, where $degf = n$ is any positive integer since M is a subspace, so by taking all linear combinations of f, one can get any polynomial with degree $n$. Is this correct, and is there a more mathematical way to express this?