First, let's expand the expression:

sin x + (cos x/2 - sin x/2)(cos x/2 + sin x/2)

= sin x + (cos^2 x/4 - sin^2 x/4)
= sin x + cos^2 x/4 - sin^2 x/4

Now we can use the Pythagorean identity sin^2 x + cos^2 x = 1 to simplify further:

sin x + (1/4 - sin^2 x/4)
= sin x + 1/4 - sin^2 x/4

Now we have the equation:
sin x + 1/4 - sin^2 x/4 = 0

Multiplying by 4 to get rid of the fraction gives us:
4sin x + 1 - sin^2 x = 0

Rearranging the terms gives:
sin^2 x + 4sin x + 1 = 0

This is a quadratic equation in terms of sin x. It can be solved using the quadratic formula.

sin x = (-b ± √(b^2 - 4ac)) / 2a
sin x = (-4 ± √(16 - 4))/2
sin x = (-4 ± √12)/2

Therefore, the solutions for sin x are:
sin x = (-4 + √12)/2 and sin x = (-4 - √12)/2

These can be further simplified if needed.