Multiply the Algebraic Expressions below:
1.) 4m² - 1                  16m² + 8m
___________   ×     ____________
18m                              6m - 3

To multiply the given algebraic expressions, we need to multiply the numerators together and the denominators together. Then, we simplify by factoring and canceling common factors.

### Given Expressions:

\[
\frac{4m^2 - 1}{18m} \times \frac{16m^2 + 8m}{6m - 3}
\]

### Step 1: Factor Each Expression

1. **Numerator of the First Expression:**
   \[
   4m^2 - 1 = (2m - 1)(2m + 1)
   \]
   (Difference of squares)

2. **Denominator of the First Expression:**
   \[
   18m = 2 \cdot 3^2 \cdot m
   \]

3. **Numerator of the Second Expression:**
   \[
   16m^2 + 8m = 8m(2m + 1)
   \]
   (Factored out \(8m\))

4. **Denominator of the Second Expression:**
   \[
   6m - 3 = 3(2m - 1)
   \]
   (Factored out 3)

### Step 2: Rewrite the Expressions

After factoring, we rewrite the expressions as:

\[
\frac{(2m - 1)(2m + 1)}{18m} \times \frac{8m(2m + 1)}{3(2m - 1)}
\]

### Step 3: Multiply the Numerators and Denominators

\[
\frac{(2m - 1)(2m + 1) \cdot 8m(2m + 1)}{18m \cdot 3(2m - 1)}
\]

### Step 4: Simplify by Canceling Common Factors

- \(2m - 1\) in the numerator and denominator.
- \(2m + 1\) in the numerator.
- \(m\) in the numerator and denominator.
- \(3\) in the denominator (\(18 = 2 \cdot 3^2\) and \(3\) factor outside)

The expression reduces to:

\[
\frac{8(2m + 1)}{18} = \frac{8(2m + 1)}{2 \cdot 9} = \frac{4(2m + 1)}{9}
\]

### Final Answer

\[
\frac{4(2m + 1)}{9}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions to Explore:
1. How do you multiply and divide algebraic fractions?
2. What is the difference of squares, and how is it applied in factoring?
3. How can we simplify complex rational expressions involving variables?
4. What are common mistakes to avoid when multiplying algebraic fractions?
5. How can we factor quadratics quickly and efficiently?

**Tip:** Always look for common factors in algebraic expressions before multiplying or dividing to simplify calculations.

Learn how to multiply algebraic expressions (4m² - 1) / (18m) × (16m² + 8m) / (6m - 3) step-by-step. This problem involves factorization of algebraic expressions and simplification of rational expressions. Suitable for Grades 9-12 students.

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