To solve this equation, we first need to rewrite it in a more manageable form.

Given equation: 2(sin^2x + sinx) = sqrt(3) + sqrt(3)sinx

We know that sin^2x = 1 - cos^2x, so we can rewrite sin^2x in terms of cosx.

Rewritten equation: 2(1 - cos^2x + sinx) = sqrt(3) + sqrt(3)sinx

Expand and simplify the equation:

2 - 2cos^2x + 2sinx = sqrt(3) + sqrt(3)sinx

Rearrange the terms:

2sinx + sqrt(3)sinx + 2 = 2cos^2x + sqrt(3)

Combine like terms:

3sinx + 2 = 2cos^2x + sqrt(3)

Since sinx = cos(90° - x), we can substitute cos^2x with sinx:

3sinx + 2 = 2sin^2x + sqrt(3)

Rearrange the equation:

0 = 2sin^2x - 3sinx + sqrt(3) - 2

Now we have a quadratic equation in terms of sinx:

2sin^2x - 3sinx + sqrt(3) - 2 = 0

To solve this equation, we can use the quadratic formula:

sinx = [3 ± sqrt(9 - 4(2)(sqrt(3) - 2))]/(4)

sinx = [3 ± sqrt(9 - 8sqrt(3) + 8)]/(4)

sinx = [3 ± sqrt(17 - 8sqrt(3))]/4

Therefore, the solutions to the equation are sinx = [3 + sqrt(17 - 8sqrt(3))]/4 and sinx = [3 - sqrt(17 - 8sqrt(3))]/4.