Let's simplify the given expression by using the trigonometric identity sin^2(x) + cos^2(x) = 1.

Given equation: sin^2(x) + 2sin(x) - 3cos^2(x) + 1 = 0

Using the trigonometric identity: 1 - cos^2(x) + 2sin(x) - 3cos^2(x) + 1 = 0
Rewrite in terms of sine and cosine: sin^2(x) + cos^2(x) + 2sin(x) - 3cos^2(x) + 1 = 0
2sin(x) - 2cos^2(x) + 1 = 0

Now we can simplify further by using the trigonometric identities sin(x) = 2sin(x)cos(x) and cos(x) = 1 - 2sin^2(x).

2(2sin(x)cos(x)) - 2(1 - 2sin^2(x))^2 + 1 = 0
4sin(x)cos(x) - 2 + 8sin^2(x) + 1 = 0
4sin(x)cos(x) + 8sin^2(x) - 1 = 0

The simplified form of the given expression is 4sin(x)cos(x) + 8sin^2(x) - 1 = 0.