Questions: Divide and simplify: (x^2+4x-5)/(x^2-6x+8) / (x+5)/(x-2)

Divide and simplify: (x^2+4x-5)/(x^2-6x+8) / (x+5)/(x-2)
Transcript text: Divide and simplify: $\frac{\frac{x^{2}+4 x-5}{x^{2}-6 x+8}}{\frac{x+5}{x-2}}$
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Solution

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Solution Steps

To divide two fractions, multiply the first fraction by the reciprocal of the second. Simplify the resulting expression by factoring and canceling common terms.

Step 1: Understand the Problem

We need to divide the fraction \(\frac{x^2 + 4x - 5}{x^2 - 6x + 8}\) by \(\frac{x + 5}{x - 2}\).

Step 2: Multiply by the Reciprocal

To divide by a fraction, multiply by its reciprocal: \[ \frac{x^2 + 4x - 5}{x^2 - 6x + 8} \times \frac{x - 2}{x + 5} \]

Step 3: Simplify the Expression

Factor the polynomials where possible:

  • \(x^2 + 4x - 5 = (x - 1)(x + 5)\)
  • \(x^2 - 6x + 8 = (x - 2)(x - 4)\)

Substitute these into the expression: \[ \frac{(x - 1)(x + 5)}{(x - 2)(x - 4)} \times \frac{x - 2}{x + 5} \]

Step 4: Cancel Common Terms

Cancel the common terms \((x + 5)\) and \((x - 2)\): \[ \frac{x - 1}{x - 4} \]

Final Answer

The simplified expression is: \[ \boxed{\frac{x - 1}{x - 4}} \]

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