To solve this equation, we will first rewrite it in terms of a single trigonometric function using the double angle formula for sine:

5sin(2x) + 5cos(x) - 8sin(x) - 4 = 0

Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:

10sin(x)cos(x) + 5cos(x) - 8sin(x) - 4 = 0

Now let's factor out the common factors:

5cos(x)(2sin(x) + 1) - 4(2sin(x) + 1) = 0

(5cos(x) - 4)(2sin(x) + 1) = 0

Now we have two possible solutions:

5cos(x) - 4 = 0 or 2sin(x) + 1 = 0

For the first solution, we have:

5cos(x) = 4
cos(x) = 4/5

Since cosine is positive in the first and fourth quadrants, the possible solutions for x are:

x = arccos(4/5) or x = 2π - arccos(4/5)

For the second solution, we have:

2sin(x) = -1
sin(x) = -1/2

Since sine is negative in the third and fourth quadrants, the possible solutions for x are:

x = -π/6 or x = -5π/6

Therefore, the solutions to the equation 5sin(2x) + 5cos(x) - 8sin(x) - 4 = 0 are:

x = arccos(4/5), 2π - arccos(4/5), -π/6, -5π/6