To simplify this expression, we can recognize that it can be rewritten as:

3sin^2x + 14sinxcosx + 8cos^2x = 0

Using the trigonometric identity sin^2x + cos^2x = 1, we can substitute cos^2x = 1 - sin^2x:

3sin^2x + 14sinx(1 - sin^2x) + 8(1 - sin^2x) = 0

Expand the expression:

3sin^2x + 14sinx - 14sin^3x + 8 - 8sin^2x = 0

Rearrange and combine like terms:

-14sin^3x + 3sin^2x - 8sinx + 8 = 0

This is a cubic equation in sin(x), and it can be solved using various methods such as factoring, graphing, or numerical methods.