Questions: (4x-20)/(x^2-x-6) ÷ (x^2-6x+5)/(x^2+x-2)

Solution Steps

To solve the given expression, we need to perform division of two rational expressions. The division of fractions is equivalent to multiplying by the reciprocal. Therefore, we will multiply the first rational expression by the reciprocal of the second. After that, we will simplify the resulting expression by factoring the polynomials and canceling out common factors.

The given expression is:

\[ \frac{4x - 20}{x^2 - x - 6} \div \frac{x^2 - 6x + 5}{x^2 + x - 2} \]

To divide by a fraction, we multiply by its reciprocal:

\[ \frac{4x - 20}{x^2 - x - 6} \times \frac{x^2 + x - 2}{x^2 - 6x + 5} \]

Factor each polynomial in the expression:

• \(4x - 20 = 4(x - 5)\)
• \(x^2 - x - 6 = (x - 3)(x + 2)\)
• \(x^2 - 6x + 5 = (x - 1)(x - 5)\)
• \(x^2 + x - 2 = (x - 1)(x + 2)\)

Substitute the factored forms into the expression:

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

Cancel out the common factors in the numerator and the denominator:

• Cancel \((x - 5)\) from the numerator and denominator.
• Cancel \((x - 1)\) from the numerator and denominator.
• Cancel \((x + 2)\) from the numerator and denominator.

The simplified expression is:

\[ \frac{4}{x - 3} \]

Final Answer