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

Verified step by step guidance

1. Factor each quadratic expression in the numerators and denominators.
2. For the first fraction's numerator $m^2+3m+2$, factor it to $(m+1)(m+2)$.
3. For the first fraction's denominator $m^2+5m+4$, factor it to $(m+1)(m+4)$.
4. For the second fraction's numerator $m^2+5m+6$, factor it to $(m+2)(m+3)$.
5. For the second fraction's denominator $m^2+10m+24$, factor it to $(m+4)(m+6)$.

Key Concepts

Polynomial Operations

Understanding how to perform operations on polynomials is crucial for solving the given problem. This includes addition, subtraction, multiplication, and division of polynomial expressions. In this case, the question involves dividing one polynomial by another, which requires knowledge of polynomial long division or synthetic division.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This is essential for simplifying the division of polynomials, as it allows for cancellation of common factors in the numerator and denominator, making the calculation more manageable.

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, multiplying, and dividing them, is key to solving the problem. This involves recognizing how to handle the division of two rational expressions effectively.