solve the recurrence relation

c[n] = -8c[n-1] -16c[n-2], c[1] =-1, c2=8
need all the work

This sounds like it's your homework or test problem, but you're
charging $10, so who knows. We will do this in the general case, and
get more specific as required. We solve the recurrence relation
c[n]=m*c[n-1]+n*c[n-2], c[1]=p, c[2]=q. We solve the equation
L^n=m*L^(n-1)+n*L^(n-2). L=0 or L^2-m*L-n=0 <=>
L=2^(-1)[(m^2+4n)^(.5)+m]. In this case, m^2+4n=0, so we get a
repeated root: L=m/2. Thus, our solution is f(x)=a*(m/2)^x+b*x*(m/2)^x
for some a and b. (If we had two distinct roots, we'd have the
equation f(x)=a*L1^x+b*L2^x, where L1 and L2 are the two roots of the
polynomial.) We use our initial conditions to solve for a and b.