Can you prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proof, but I think it is better to broaden the scope for this bounty question.

A psd matrix of the form $\left(\begin{smallmatrix} a & b & c\\ b & c & d \\ c & d & e \end{smallmatrix}\right)$ can be written as the sum of finitely many rank 1 matrices of the form $\left(\begin{smallmatrix} x^4 & x^3 y & x^2 y^2\\ x^3 y & x^2 y^2 & x y^3 \\ x^2 y^2 & x y^3 & y^4 \end{smallmatrix}\right)$?

**Edit** Thank you Greg for your answer. In our comments, we observe that every psd matrix of the form $\left(\begin{smallmatrix}
a & b & c & d\\
b & c & d & e\\
c & d & e & f\\
d & e & f & g
\end{smallmatrix}\right)$ is a finite sum of rank 1 and rank 2 matrices. Can one prove the following in a convex analytic manner?

A psd matrix of the form $\left(\begin{smallmatrix} a & b & c & d\\ b & c & d & e\\ c & d & e & f\\ d & e & f & g \end{smallmatrix}\right)$ can be written as the sum of finitely many rank 1 matrices of the form $\left(\begin{smallmatrix} x^6 & x^5 y & x^4 y^2 & x^3 y^3\\ x^5 y & x^4 y^2 & x^3 y^3 & x^2 y^4\\ x^4 y^2 & x^3 y^3 & x^2y^4 & x y^5 \\ x^3y^3 & x^2y^4 & x y^5 & y^6 \end{smallmatrix}\right)$?

** Motivation ** The first question is a convex analytic proof to the (known fact) that $P_{2,4}=\Sigma_{2,4}$, while the second question is to prove $P_{2,6}=\Sigma_{2,6}$. Below, I describe the origin of the problem for the former.

I came upon this problem as a convex analytic approach to the (known fact) that $P_{2,4}=\Sigma_{2,4}$. Here $P_{2,4}$ is the cone of nonnegative-valued binary quartics. $\Sigma_{2,4}$ is the cone of binary quartics that are sums of squares. (A binary quartic is a homogeneous polynomial in 2 variables of degree 4). Obviously $P_{2,4}\subseteq \Sigma_{2,4}$. The standard proof that equality holds is by dehomonogizing and applying the Fundamental theorem of algebra.

Since the cone $\Sigma_{2,4}$ is closed in the vector space ${\mathbb{R}}[x,y]_4$ of homogeneous polynomials in 2 variables of degree 4, a separation theorem in convex geometry provides a necessary and sufficient condition for a binary quartic $f$ to lie in $\Sigma_{2,4}$. This condition is that, for any linear functional $T$ which spans an extremal ray of the dual cone $\Sigma_{2,4}^{\vee}$, we have $T(f)\ge 0$.

The cone of positive semidefinite matrices of the form $\left(\begin{smallmatrix} a & b & c\\ b & c & d \\ c & d & e \end{smallmatrix}\right)$ is isomorphic to $\Sigma_{2,4}^{\vee}$. Under this isomorphism, point evaluations correspond to rank 1 matrices of the form $\left(\begin{smallmatrix} x^4 & x^3 y & x^2 y^2\\ x^3 y & x^2 y^2 & x y^3 \\ x^2 y^2 & x y^3 & y^4 \end{smallmatrix}\right)$. The above proof of which I'm seeking a convex analytic proof is equivalent to the assertion $P_{2,4}=\Sigma_{2,4}$.