Sin^2x-9sinxcosx+3cos^2x=-1

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To simplify the given expression, we will use the trigonometric identities:

1) sin^2x + cos^2x = 1
2) sin2x = 2sinxcosx

Starting with the given expression:
sin^2x - 9sinxcosx + 3cos^2x = -1

Rearranging terms we get:
(sin^2x + 3cos^2x) - 9sinxcosx = -1

Now, substitute sin^2x + cos^2x = 1:
1 + 2cos^2x - 9sinxcosx = -1

Rearranging terms:
2cos^2x - 9sinxcosx = -2

Applying double angle identity for cosine:
2(1 - 2sin^2x) - 9sinxcosx = -2
2 - 4sin^2x - 9sinxcosx = -2

Rearranging terms one more time:
4sin^2x + 9sinxcosx = 4

Now, recall that sin2x = 2sinxcosx:
2(2sin^2x + 9sinxcosx) = 2(4)
4sin2x = 8

Therefore, the simplified form of the given expression is 4sin2x = 8.