Multiply or divide, as indicated. See Example 3. ((m² + 3m + 2) / (m² + 5m + 4)) ÷ ((m² + 5m + 6) / (m² + 10m + 24))

Verified step by step guidance

1. Rewrite the expression as a division of two rational expressions: \(\frac{m^{2} + 3m + 2}{m^{2} + 5m + 4} \div \frac{m^{2} + 5m + 6}{m^{2} + 10m + 24}\).
2. Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, rewrite the expression as \(\frac{m^{2} + 3m + 2}{m^{2} + 5m + 4} \times \frac{m^{2} + 10m + 24}{m^{2} + 5m + 6}\).
3. Factor each quadratic polynomial completely: - \(m^{2} + 3m + 2\) factors to \((m + 1)(m + 2)\), - \(m^{2} + 5m + 4\) factors to \((m + 1)(m + 4)\), - \(m^{2} + 5m + 6\) factors to \((m + 2)(m + 3)\), - \(m^{2} + 10m + 24\) factors to \((m + 4)(m + 6)\).
4. Substitute the factored forms back into the expression: \(\frac{(m + 1)(m + 2)}{(m + 1)(m + 4)} \times \frac{(m + 4)(m + 6)}{(m + 2)(m + 3)}\).
5. Cancel out common factors in numerator and denominator across the multiplication, then multiply the remaining factors to write the simplified expression.

Key Concepts

Factoring Quadratic Expressions

Factoring involves rewriting quadratic expressions as products of binomials. This simplifies complex rational expressions by breaking down polynomials into simpler factors, making multiplication or division easier. Recognizing common patterns like trinomials and difference of squares is essential.

Dividing Rational Expressions

Dividing rational expressions requires multiplying by the reciprocal of the divisor. This means flipping the second fraction and then multiplying numerators and denominators. Simplifying before multiplying helps reduce complexity and avoid errors.

Simplifying Rational Expressions

Simplifying rational expressions involves canceling common factors in the numerator and denominator after factoring. This reduces the expression to its simplest form, making it easier to interpret and solve. Always factor completely before canceling.