Multiply or divide, as indicated. See Example 3. ((4a + 12) / (2a - 10)) ÷ ((a² - 9) / (a² - a - 20))

Verified step by step guidance

1. Rewrite the given expression as a division of two rational expressions: \(\frac{4a + 12a^{2} - 9}{2a} \div \frac{10a^{2} - a - 20}{1}\).
2. Recall that dividing by a fraction is the same as multiplying by its reciprocal, so rewrite the expression as \(\frac{4a + 12a^{2} - 9}{2a} \times \frac{1}{10a^{2} - a - 20}\).
3. Factor all polynomials in the numerators and denominators where possible. For example, factor \$4a + 12a^{2} - 9\(, \)10a^{2} - a - 20$, and any other polynomial expressions.
4. After factoring, rewrite the expression with factored forms and then cancel out any common factors between numerators and denominators.
5. Multiply the remaining numerators together and the remaining denominators together to write the simplified expression.

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of simpler polynomials or factors. This is essential for simplifying expressions, especially when multiplying or dividing polynomials, as it helps identify common factors that can be canceled out.

Division of Rational Expressions

Dividing rational expressions requires multiplying the first expression by the reciprocal of the second. This process involves flipping the numerator and denominator of the divisor and then performing multiplication, followed by simplification.

Simplification of Algebraic Fractions

Simplifying algebraic fractions means reducing them to their simplest form by canceling common factors in the numerator and denominator. This step is crucial after multiplication or division to present the expression in a clear and concise form.