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