To solve this trigonometric equation, we can first rewrite it using the double angle formula for sine, which states that sin(2x) = 2sin(x)cos(x).

So, we have:

2sin²x + 1.5(2sin(x)cos(x)) - 3cos²x = 1
2sin²x + 3sin(x)cos(x) - 3cos²x = 1

Now, we can use the Pythagorean identity sin²x + cos²x = 1 to substitute for one of the trigonometric functions. Let's substitute for sin²x:

2(1-cos²x) + 3sin(x)cos(x) - 3cos²x = 1
2 - 2cos²x + 3sin(x)cos(x) - 3cos²x = 1
-5cos²x + 3sin(x)cos(x) - 1 = 0

Now, we can rewrite the equation in terms of only cos(x):

5cos²x - 3sin(x)cos(x) + 1 = 0

Solving this quadratic equation in cos(x) might be a bit tricky. We might need to use the quadratic formula:

cos(x) = [3sin(x) ± sqrt((3sin(x))² - 451)] / 10

We can then use the Pythagorean identity sin²x + cos²x = 1 to find the corresponding sine values for the solutions of cos(x).

I hope this helps! Let me know if you have any questions.