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 $$L(\frac{d}{dt})y=t^m e^{\gamma t}, \ \gamma \in \mathbb C,$$ 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 $y=Q(t)e^{\gamma t}$ where $Q$ is a polynomial that satisfies the following ode with constant coefficients $$\frac{p^{(n)}(\gamma)}{n!}Q^{(n)} + \cdots + \frac{p'(\gamma)}{1!}Q' + \frac{p(\gamma)}{0!}Q= t^m,$$ where $p=p(\lambda)$ is the characteristic polynomial associated to $L$. The short and very elementary proof of this formula, with some examples, is over there. Using it, we find a particular solution through solving a standard linear system.
The same formula shows that if $a$ is a root of multiplicity $k$ of $p(\lambda)=0$ then $e^{at},\ldots, t^{k-1}e^{at}$ are solutions of the homogeneous equation $Ly=0$.
Oswaldo R. B. de Oliveira