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

Question

UpStudy AI · Accepted Answer

To simplify the given expression:

$$
\frac{15 - 14x - 8x^2}{4x^2 + 4x - 15} \div \frac{4x^2 + 13x - 12}{3x^2 + 13x + 4}
$$

we can follow these steps:

1. **Rewrite the Division as Multiplication:**

$$
\frac{15 - 14x - 8x^2}{4x^2 + 4x - 15} \times \frac{3x^2 + 13x + 4}{4x^2 + 13x - 12}
$$

2. **Factor Each Quadratic Expression:**

- **Factor the numerator $$15 - 14x - 8x^2$$:**
     $$
     -8x^2 - 14x + 15 = -1(8x^2 + 14x - 15) = -1(4x - 3)(2x + 5)
     $$

- **Factor the denominator $$4x^2 + 4x - 15$$:**
     $$
     4x^2 + 4x - 15 = (2x - 3)(2x + 5)
     $$

- **Factor the numerator $$3x^2 + 13x + 4$$:**
     $$
     3x^2 + 13x + 4 = (3x + 1)(x + 4)
     $$

- **Factor the denominator $$4x^2 + 13x - 12$$:**
     $$
     4x^2 + 13x - 12 = (4x - 3)(x + 4)
     $$

3. **Substitute the Factored Forms:**

$$
\frac{-1(4x - 3)(2x + 5)}{(2x - 3)(2x + 5)} \times \frac{(3x + 1)(x + 4)}{(4x - 3)(x + 4)}
$$

4. **Cancel Out Common Factors:**

- $$4x - 3$$ cancels out.
- $$2x + 5$$ cancels out.
- $$x + 4$$ cancels out.

5. **Simplify the Remaining Expression:**

$$
-1 \times \frac{3x + 1}{2x - 3} = -\frac{3x + 1}{2x - 3}
$$

**Final Answer:**

$$
-\frac{3x + 1}{2x - 3}
$$
$$ -\frac{3x + 1}{2x - 3} $$

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