To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, follow these steps:
- Set up the long division: Write \( 15x^2 + 14x - 8 \) under the division symbol and \( 3x + 4 \) outside.
\[ \begin{array}{r|r} 3x + 4 & 15x^2 + 14x - 8 \ \end{array} \]
- Divide the leading term: Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):
\[ \frac{15x^2}{3x} = 5x \]
- Multiply and subtract: Multiply \( 5x \) by \( 3x + 4 \) and subtract the result from the original polynomial.
\[ 5x(3x + 4) = 15x^2 + 20x \]
Now subtract:
\[ 15x^2 + 14x - 8 - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8 \]
- Repeat the process: Now use \( -6x - 8 \) as the new dividend. Divide the leading term \( -6x \) by \( 3x \):
\[ \frac{-6x}{3x} = -2 \]
- Multiply and subtract again: Multiply \( -2 \) by \( 3x + 4 \) and subtract:
\[ -2(3x + 4) = -6x - 8 \]
Subtract:
\[ -6x - 8 - (-6x - 8) = 0 \]
- Final quotient: Since there is no remainder, we conclude the division here.
The quotient from the long division of \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) is:
\[ \boxed{5x - 2} \]