To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, let's go through the steps:
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Setup the long division: \[ \text{Divide: } 15x^2 + 14x - 8 \div (3x + 4) \]
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Divide the leading term: Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \): \[ \frac{15x^2}{3x} = 5x \]
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Multiply and subtract: Multiply \( 5x \) by \( 3x + 4 \): \[ 5x(3x + 4) = 15x^2 + 20x \]
Subtract this product from the original polynomial: \[ (15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8 \]
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Repeat the process: Now take the leading term \(-6x\) and divide it by \(3x\): \[ \frac{-6x}{3x} = -2 \]
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Multiply and subtract again: Multiply \(-2\) by \(3x + 4\): \[ -2(3x + 4) = -6x - 8 \]
Subtract this from \(-6x - 8\): \[ (-6x - 8) - (-6x - 8) = 0 \]
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Conclusion: The result of the division is: \[ 5x - 2 \] with a remainder of 0. Thus, the quotient of the division \( 15x^2 + 14x - 8 \div (3x + 4) \) is: \[ \boxed{5x - 2} \]