Recurrence Substitution Method with multiple givens
Let $S(n) = T(n) + 1.5$. Then
$$
S(n) = T(n) + 1.5 = T(n-1) + 2T(n-2) + 4.5 = S(n-1) + 2S(n-2).
$$
The recurrence $S(n) = S(n-1) + 2S(n-2)$ is homogeneous with constant coefficients, so we know how to solve it. We first solve the equation $\lambda^2 = \lambda + 2$, finding that the solutions are $2,-1$. This implies that there exist constants $A,B$ such that
$$
S(n) = A 2^n + B (-1)^n.
$$
Since $S(0) = 4.5$ and $S(1) = 6.5$, we see that $A + B = 4.5$ and $2A - B = 6.5$, from which we find that $A = 11/3$ and $B = 5/6$. Therefore
$$
T(n) = \frac{11}{3} 2^n + \frac{5}{6} (-1)^n - \frac{3}{2}.
$$