84. \( \frac{15-14 x-8 x^{2}}{4 x^{2}+4 x-15} \div \frac{4 x^{2}+13 x-12}{3 x^{2}+13 x+4} \)

To solve the expression \( \frac{15-14 x-8 x^{2}}{4 x^{2}+4 x-15} \div \frac{4 x^{2}+13 x-12}{3 x^{2}+13 x+4} \), we begin by rewriting the division as multiplication by the reciprocal: \[ \frac{15-14 x-8 x^{2}}{4 x^{2}+4 x-15} \cdot \frac{3 x^{2}+13 x+4}{4 x^{2}+13 x-12} \] Now we will proceed by factoring each polynomial involved in the expression. 1. **Factoring \(15 - 14x - 8x^2\)**: \[ -8x^2 - 14x + 15 = -1(8x^2 + 14x - 15) \] We search for two numbers that multiply to \( -8 \cdot 15 = -120\) and add to \( 14\). Those numbers are \( 20 \) and \( -6 \). Thus, we can rewrite the quadratic: \[ 8x^2 + 20x - 6x - 15 = 4x(2x + 5) - 3(2x + 5) = (4x - 3)(2x + 5) \] Therefore, \[ 15 - 14x - 8x^2 = -(4x - 3)(2x + 5) \] 2. **Factoring \(4x^2 + 4x - 15\)**: We look for two numbers that multiply to \( 4 \cdot (-15) = -60 \) and add to \( 4 \). The numbers \( 10 \) and \( -6 \) work: \[ 4x^2 + 10x - 6x - 15 = 2x(2x + 5) - 3(2x + 5) = (2x - 3)(2x + 5) \] 3. **Factoring \(4x^2 + 13x - 12\)**: For this, we find numbers that multiply to \( 4 \cdot (-12) = -48 \) and sum to \( 13 \). The numbers are \( 16 \) and \( -3 \): \[ 4x^2 + 16x - 3x - 12 = 4x(x + 4) - 3(x + 4) = (4x - 3)(x + 4) \] 4. **Factoring \(3x^2 + 13x + 4\)**: This time, we look for two numbers that multiply to \( 3 \cdot 4 = 12 \) and add to \( 13 \). Those numbers are \( 12 \) and \( 1 \): \[ 3x^2 + 12x + x + 4 = 3x(x + 4) + 1(x + 4) = (3x + 1)(x + 4) \] Now, substituting the factored forms into the expression, we get: \[ \frac{-(4x - 3)(2x + 5)}{(2x - 3)(2x + 5)} \cdot \frac{(3x + 1)(x + 4)}{(4x - 3)(x + 4)} \] Next, we can cancel out common factors: - \(2x + 5\) cancels with \( 2x + 5\) - \(4x - 3\) cancels with \( 4x - 3\) - \(x + 4\) cancels with \( x + 4\) This leaves us with: \[ \frac{-(3x + 1)}{(2x - 3)} \] Thus, the final expression simplifies to: \[ \frac{-(3x + 1)}{(2x - 3)} \] This is the simplified result of the given expression.