To simplify this expression, we can use the angle addition formulas for sine and cosine.

sin(x) + sin(5x) + cos(x) + cos(5x) = 2sin(3x)cos(2x) + 2cos(3x)cos(2x)

= 2(cos(3x) + sin(3x))cos(2x)

= 2sqrt(2)cos(2x + pi/4)

Therefore, the expression simplifies to:

2sqrt(2)cos(2x + pi/4) = 0

To find the solutions for x, we can set cos(2x + pi/4) = 0:

2x + pi/4 = (2n + 1)pi/2, where n is an integer.

2x = (2n + 1)pi/2 - pi/4

x = (4n + 1)pi/4 - pi/8

Therefore, the general solution for x is:

x = (4n + 1)pi/4 - pi/8, where n is an integer.