Questions: (5 x^2+5 x-100)/(x^5-2 x^4-8 x^3) * (x^2-4)/(15 x-30)

Solution Steps

To solve the given expression, we need to simplify the fractions and then multiply them. Here are the high-level steps:

1. Factorize the numerators and denominators of both fractions.
2. Simplify the fractions by canceling out common factors.
3. Multiply the simplified fractions together.
4. Write the result as a single fraction.

We start by factoring the numerators and denominators of the given fractions:

1. The numerator of the first fraction: \[ 5x^2 + 5x - 100 = 5(x - 4)(x + 5) \]

2. The denominator of the first fraction: \[ x^5 - 2x^4 - 8x^3 = x^3(x - 4)(x + 2) \]

3. The numerator of the second fraction: \[ x^2 - 4 = (x - 2)(x + 2) \]

4. The denominator of the second fraction: \[ 15x - 30 = 15(x - 2) \]

Now we can simplify each fraction:

1. The first fraction simplifies to: \[ \frac{5(x - 4)(x + 5)}{x^3(x - 4)(x + 2)} = \frac{5(x + 5)}{x^3(x + 2)} \]

2. The second fraction simplifies to: \[ \frac{(x - 2)(x + 2)}{15(x - 2)} = \frac{x + 2}{15} \]

Next, we multiply the simplified fractions together: \[ \frac{5(x + 5)}{x^3(x + 2)} \cdot \frac{x + 2}{15} = \frac{5(x + 5)(x + 2)}{15x^3(x + 2)} \]

We can cancel the common factor \((x + 2)\): \[ \frac{5(x + 5)}{15x^3} = \frac{x + 5}{3x^3} \]

Final Answer