86. \( \frac{10+7 x-12 x^{2}}{8 x^{2}-2 x-15} \div \frac{6 x^{2}-13 x+5}{10 x^{2}-13 x+4} \)

To simplify the expression \[ \frac{10 + 7x - 12x^2}{8x^2 - 2x - 15} \div \frac{6x^2 - 13x + 5}{10x^2 - 13x + 4}, \] we will rewrite the division as multiplication by the reciprocal: \[ \frac{10 + 7x - 12x^2}{8x^2 - 2x - 15} \cdot \frac{10x^2 - 13x + 4}{6x^2 - 13x + 5}. \] Next, let's factor each polynomial. **Step 1: Factor each polynomial.** 1. For \(10 + 7x - 12x^2\): \[ -12x^2 + 7x + 10 \quad \text{(reorder)} \] We can factor this using the quadratic formula where \(a = -12\), \(b = 7\), \(c = 10\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{7^2 - 4(-12)(10)}}{2(-12)} = \frac{-7 \pm \sqrt{49 + 480}}{-24} = \frac{-7 \pm \sqrt{529}}{-24}. \] Since \(\sqrt{529} = 23\), \[ x = \frac{-7 + 23}{-24} = \frac{16}{-24} = -\frac{2}{3} \quad \text{and} \quad x = \frac{-7 - 23}{-24} = \frac{-30}{-24} = \frac{5}{4}. \] Thus, we can write: \[ 10 + 7x - 12x^2 = -12(x + \frac{2}{3})(x - \frac{5}{4}). \] 2. For \(8x^2 - 2x - 15\): We can factor by inspection: \[ 8x^2 - 2x - 15 = (4x + 5)(2x - 3). \] 3. **The reciprocal** \(10x^2 - 13x + 4\): This factors as: \[ (5x - 4)(2x - 1). \] 4. For \(6x^2 - 13x + 5\): This factors as: \[ (2x - 5)(3x - 1). \] **Step 2: Rewrite the expression with the factors.** Now substituting the factors into the expression we have: \[ \frac{-12(x + \frac{2}{3})(x - \frac{5}{4})}{(4x + 5)(2x - 3)} \cdot \frac{(5x - 4)(2x - 1)}{(2x - 5)(3x - 1)}. \] **Step 3: Cancel common factors.** We see that: - \((2x - 5)\) can be canceled with \((2x - 5)\). After simplification, we combine and rearrange the terms. After grouping, cancellation can lead to a further reduced expression. The final simplified result (after canceling common factors) is: \[ \frac{-12(5x - 4)(x + \frac{2}{3})(x - \frac{5}{4})}{(4x + 5)(2x - 3)(3x - 1)}. \] This completes the simplification of the original expression.