To simplify the given expression
[tex]\[
\frac{x^2-12 x+32}{2 x^3-18 x^2+40 x} \cdot \frac{x^2-1}{x^2-7 x-8} \div \frac{x^2+4 x-5}{25-x^2}
\][/tex]
we will proceed step-by-step, first by simplifying each component individually and then combining the results.
### Step 1: Simplify each individual fraction
1. Simplify [tex]\(\frac{x^2-12x+32}{2x^3-18x^2+40x}\)[/tex]
- Factor the numerator: [tex]\(x^2 - 12x + 32 = (x - 4)(x - 8)\)[/tex]
- Factor the denominator: [tex]\(2x^3 - 18x^2 + 40x = 2x(x^2 - 9x + 20) = 2x(x-4)(x-5)\)[/tex]
- Simplified form: [tex]\(\frac{(x-4)(x-8)}{2x(x-4)(x-5)} = \frac{x-8}{2x(x-5)}\)[/tex] (Cancel out [tex]\(x-4\)[/tex])
2. Simplify [tex]\(\frac{x^2-1}{x^2-7x-8}\)[/tex]
- Factor the numerator: [tex]\(x^2 - 1 = (x-1)(x+1)\)[/tex]
- Factor the denominator: [tex]\(x^2 - 7x - 8 = (x-8)(x+1)\)[/tex]
- Simplified form: [tex]\(\frac{(x-1)(x+1)}{(x-8)(x+1)} = \frac{x-1}{x-8}\)[/tex] (Cancel out [tex]\(x+1\)[/tex])
3. Simplify [tex]\(\frac{x^2+4x-5}{25-x^2}\)[/tex]
- Factor the numerator: [tex]\(x^2 + 4x - 5 = (x+5)(x-1)\)[/tex]
- Factor the denominator: [tex]\(25 - x^2 = (5-x)(5+x) = -(x-5)(x+5)\)[/tex]
- Simplified form: [tex]\(\frac{(x+5)(x-1)}{-(x-5)(x+5)} = \frac{x-1}{-(x-5)} = -\frac{x-1}{x-5}\)[/tex] (Cancel out [tex]\(x+5\)[/tex])
### Step 2: Combine the simplified components
Substitute the simplified forms into the original expression:
[tex]\[
\frac{x-8}{2x(x-5)} \cdot \frac{x-1}{x-8} \div \left(-\frac{x-1}{x-5}\right)
\][/tex]
### Step 3: Perform the operations
1. Multiply the first two fractions:
[tex]\[
\frac{x-8}{2x(x-5)} \cdot \frac{x-1}{x-8} = \frac{(x-8)(x-1)}{2x(x-5)(x-8)} = \frac{x-1}{2x(x-5)} \quad \text{(Cancel out \(x-8\))}
\][/tex]
2. Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{x-1}{2x(x-5)} \div \left(-\frac{x-1}{x-5}\right) = \frac{x-1}{2x(x-5)} \cdot \frac{x-5}{-(x-1)} = \frac{(x-1)(x-5)}{2x(x-5)} \cdot \frac{1}{-(x-1)} = \frac{1}{2x} \cdot \frac{1}{-1} = -\frac{1}{2x}
\][/tex]
### Final Simplification:
Therefore, the fully simplified form of the given expression is:
[tex]\[
-\frac{1}{2x}
\][/tex]
The final simplified result is:
[tex]\[
-\frac{1}{2x}
\][/tex]
