To solve this equation, we can rewrite it in terms of sine and cosine:

3 sin^2$x$ - 3 sin$x$ cos$x$ - 4 cos^2$x$ = -2

Using the trigonometric identity sin^2$x$ + cos^2$x$ = 1, we can rewrite the equation as:

3$1-co{s}^2(x)$ - 3 sin$x$ cos$x$ - 4 cos^2$x$ = -2

Expanding and simplifying the equation gives:

3 - 3cos^2$x$ - 3 sin$x$ cos$x$ - 4 cos^2$x$ = -2

Rearranging the terms and combining like terms gives:

-7cos^2$x$ - 3 sin$x$ cos$x$ = -5

Now we can use the trigonometric identity sin$2x$ = 2sin$x$cos$x$ to simplify the equation further:

-7cos^2$x$ - 3 sin$x$ cos$x$ = -5
-7cos^2$x$ - 3 sin$2x$ = -5

Now we can solve for cos$x$ using the Pythagorean identity cos^2$x$ + sin^2$x$ = 1:

-7$1-si{n}^2(x)$ - 3 sin$2x$ = -5
-7 + 7sin^2$x$ - 3 sin$2x$ = -5

Substitute sin$2x$ = 2sin$x$cos$x$:

-7 + 7sin^2$x$ - 6sin$x$cos$x$ = -5

Rearrange the equation to solve for sin$x$ and cos$x$:

7sin^2$x$ - 6sin$x$cos$x$ = 2

Now we need additional information or constraints to fully solve this equation for sin$x$ and cos$x$.