Hi Ryan and others,
Besides the very beautiful proof by Tao, a very nice and easy linear algebra approach to the undetermined coefficients method can be found in C. C. Ross ``Why the method of undetermined coefficients works'', Am. Math. Monthly 98 (1991), pp. 747-749.
I also suggest R. C. Gupta ``Linear differential equations with constant coefficients: a recursive alternative to the method of the undetermined coefficients'', Int. J. Math. Educ. Sci. Technol. 27 (1996), pp. 757-760.
A very easy formula (quite good for the students) that gives a particular solution of the equation $$L(\frac{d}{dt})y=t^m e^{q t}, \ q \in \mathbb C,$$ where $L$ is a nth order linear differential operator with constant coefficients, can be found in de Oliveira ``A formula substituting the undetermined coefficients and the annihilator methods'', Int. J. Math. Educ. Sci. Technol. 44 (3), 2013, pp. 462-468. The proof of the formula is in p. 465 and it is short and very elementary.
The idea is the following: the given ode has a particular solution $y=f(t)e^{q t}$ where $f$ is a polynomial that satisfies the following ode with constant coefficients $$\frac{p^{(n)}(q)}{n!}f^{(n)} + \cdots + \frac{p'(q)}{1!}f' + \frac{p(q)}{0!}f= t^m,$$ where $p=p(\lambda)$ is the characteristic polynomial associated to $L$. The short and elementary proof of this formula, with some examples, is over there. Using it, we find a particular solution through solving a triangular linear system.
The same formula shows that if $q$ is a root of multiplicity $k$ of $p(\lambda)=0$ then $e^{qt},\ldots, t^{k-1}e^{qt}$ are solutions of the homogeneous equation $Ly=0$, since we have $p(q)=\cdots=p^{(k-1)}(q)=0$ and $\frac{d^l}{dt^l}(t^j)=0$ if $l>j$. Furthermore, in this case the formula boils down to $$\frac{p^{(n)}(q)}{n!}f^{(n)} + \cdots + \frac{p^{(k)}(q)}{k!}f^{(k)} = t^m,$$ which shows that $f^{(k)}$ is a polynomial of order $m$. Integrating it $k$-times we are able to choose $f(t)=t^kg(t)$, where $g$ is a polynomial of order $m$.
Oswaldo R. B. de Oliveira