q'' + 4q' +3q
the method of undetermined coefficients is used when you have a driving function.
If you just want the homogeneous solution the method doesn't enter into it.
Just find the roots $r_1, r_2$ of the characteristic equation
$s^2 + 4s + 3=0$
and the solutions will be
$q(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x},~c_1, c_2 \in \mathbb{C}$
In this case
$(s+3)(s+1)=0$
$r_1 = -3,~r_2 = -1$
$q(x) = c_1 e^{-3x}+c_2 e^{-x}, c_2 \in \mathbb{C}$