15) \( \frac{8 x^{3}-3 x-6 x^{4}-2 x^{2}-1}{3 x+4} \)

To simplify the expression \( \frac{8 x^{3}-3 x-6 x^{4}-2 x^{2}-1}{3 x+4} \), start by rearranging the numerator in standard form. The numerator can be rewritten as \( -6x^4 + 8x^3 - 2x^2 - 3x - 1 \). Now you can use polynomial long division to divide this polynomial by \( 3x + 4 \). First, divide the leading term of the numerator, \( -6x^4 \), by the leading term of the denominator, \( 3x \), which gives you \( -2x^3 \). Multiply \( -2x^3 \) by \( 3x + 4 \) to get \( -6x^4 - 8x^3 \). Subtract this from the original polynomial: \[ (-6x^4 + 8x^3) - (-6x^4 - 8x^3) = 16x^3 \] Bring down the next term: \[ 16x^3 - 2x^2 = (16x^3 - 2x^2) \rightarrow \text{Divide again: } \frac{16x^3}{3x} = \frac{16}{3}x^2 \] Continue this process until you've processed all terms. Eventually, you will reach a remainder, which can be combined with the polynomial division result to express the final simplified form. This long division will yield a quotient in terms of powers of \( x \) and a remainder that consists of a degree less than the denominator, providing you with a complete and simplified fraction. If you continue through these steps carefully, you'll arrive at the final expression of the function in its simplest form. Happy dividing!