The characteristic polynomial is \(x^2 - 6x + 9\text{.}\) We solve the characteristic equation

\begin{equation*} x^2 - 6x + 9 = 0 \end{equation*}

by factoring:

\begin{equation*} (x - 3)^2 = 0 \end{equation*}

so \(x =3\) is the only characteristic root. Therefore we know that the solution to the recurrence relation has the form

\begin{equation*} a_n = a 3^n + bn3^n \end{equation*}

for some constants \(a\) and \(b\text{.}\) Now use the initial conditions:

\begin{align*} a_0 = 1 \amp = a 3^0 + b\cdot 0 \cdot 3^0 = a\\ a_1 = 4 \amp = a\cdot 3 + b\cdot 1 \cdot3 = 3a + 3b\text{.} \end{align*}

Since \(a = 1\text{,}\) we find that \(b = \frac{1}{3}\text{.}\) Therefore the solution to the recurrence relation is

\begin{equation*} a_n = 3^n + \frac{1}{3}n3^n\text{.} \end{equation*}