Multiply or divide as indicated. $\frac{6x + 9}{3x - 15} \cdot \frac{x - 5}{4x + 6}$

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{6x+9}{3x-15} \cdot \frac{x-5}{4x+6}\).
2. Factor all polynomials in the numerators and denominators where possible. For example, factor out the greatest common factor (GCF) from each: \$6x+9 = 3(2x+3)\(, \)3x-15 = 3(x-5)\(, and \)4x+6 = 2(2x+3)$.
3. Rewrite the expression using the factored forms: \(\frac{3(2x+3)}{3(x-5)} \cdot \frac{x-5}{2(2x+3)}\).
4. Look for common factors in the numerators and denominators across the entire expression that can be canceled out. Cancel these common factors to simplify the expression.
5. After canceling, multiply the remaining numerators together and the remaining denominators together to write the simplified product as a single fraction.

Key Concepts

Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler expressions. This is essential for simplifying rational expressions by canceling common factors. For example, 6x + 9 can be factored as 3(2x + 3), and 3x - 15 as 3(x - 5).

Multiplication of Rational Expressions

Multiplying rational expressions involves multiplying the numerators together and the denominators together. After multiplication, the expression should be simplified by factoring and canceling common factors to reduce it to simplest form.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their simplest form by canceling common factors in the numerator and denominator. This requires identifying and factoring expressions fully to see which terms can be canceled.