Given this recurrence equation:
$u_1 = 0, u_2 = 1$
$u_n = 4u_{n−1} + 4u_{n−2}$
Is the correct characteristic equation:
$4x+4 = 4$
EDIT:
Complete solve:
The characteristic equation is
$x^2-4x-4=0$
We solve the quadratic equation...
$\alpha = 5$
$\beta=-1$
So:
$u_n = c_1 \alpha^n + c_2 \beta^n$
We solve the equation...
$c_1 = 1/30$
$c_2 = 1/6$
Finally:
$u_n = \dfrac{5^n}{30} + \dfrac{(-1)^n}{6}$