Show that for each polynomial $q\in \mathcal{P}(\mathcal{R})$, there exists a polynomial $p \in \mathcal{P}(\mathcal{R})$ with $((x^2 + 5x + 7)p)'' = q$
Where $\mathcal{P}_{m}(\mathcal{R})$ = Set of polynomials in the reals of degree $\le m$
Relevant Theorem: Suppose $V$ is finite dimensional and $T \in \mathcal{L}(V,V)$. I.e T is a linear operator. Then the following are equivalent: a) $T$ is invertible b) $T$ is injective c) $T$ is surjective
My question pertains to how to draw the conclusion for what we are attempting to achieve at the end step. So going through the mechanics we would define a map $T: \mathcal{P}_{m}(\mathcal{R}) \rightarrow \mathcal{P}_{m}(\mathcal{R})$.
From here we would establish that $T$ is injective, which then implies $T$ is surjective and as a consequence $T$ is invertible.
So my question is: Since we have established $T$ to be invertible does the existence of the polynomial $p$ arise because we would be able to derive said polynomial $p$ from the invertible map on which we could apply the polynomial $q$ ?
TL:DR: There exists the map $T^{-1}$ on which $T^{-1}(q) = p$
I'm assuming that $\mathcal{P}(\mathbb{R})$ is the space of all real polynomials, and $\mathcal{P}_m(\mathbb{R})$ is the space of all real polynomials of degree $\le m$.
Define $T : \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ as $Tp = ((x^2 + 5x + 7)p)''$ and show that $T$ is injective (hint: $\deg Tp = \deg p$ for all $p \ne 0$).
If $\deg q = m$, consider the restriction $T|_{\mathcal{P}_m(\mathbb{R})} : \mathcal{P}_m(\mathbb{R}) \to \mathcal{P}_m(\mathbb{R})$. It is well-defined because again$\deg Tp = \deg p$ for all $p \ne 0$.
Since $T$ is injective, $T|_{\mathcal{P}_m(\mathbb{R})}$ is also injective. The space $\mathcal{P}_m(\mathbb{R})$ is finite-dimensional (namely, $\dim \mathcal{P}_m(\mathbb{R}) = m+1$) so $T|_{\mathcal{P}_m(\mathbb{R})}$ is also surjective. Therefore there exists $p \in \mathcal{P}_m(\mathbb{R})$ such that $T|_{\mathcal{P}_m(\mathbb{R})}p = q$, meaning $Tp = q$.
Also have a look at this theorem.