Examples of multiplying algebraic fractions with various expressions and variables.

Problem: Solve the given examples involving multiplication of algebraic expressions.

We will solve each example step by step. The general approach involves:
1. Multiplying the numerators together.
2. Multiplying the denominators together.
3. Simplifying the resulting expression by canceling out common factors in the numerator and denominator.

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#### Example 1:
$$
\frac{x}{3} \times x
$$

- Rewrite \( x \) as \( \frac{x}{1} \):
$$
\frac{x}{3} \times \frac{x}{1}
$$

- Multiply the numerators and denominators:
$$
\frac{x \cdot x}{3 \cdot 1} = \frac{x^2}{3}
$$

- Final Answer:
$$
\boxed{\frac{x^2}{3}}
$$

---

#### Example 2:
$$
\frac{m}{4b} \times \frac{2a}{5}
$$

- Multiply the numerators and denominators:
$$
\frac{m \cdot 2a}{4b \cdot 5} = \frac{2am}{20b}
$$

- Simplify by canceling the common factor of 2 in the numerator and denominator:
$$
\frac{2am}{20b} = \frac{am}{10b}
$$

- Final Answer:
$$
\boxed{\frac{am}{10b}}
$$

---

#### Example 3:
$$
\frac{y}{y+3} \times \frac{y^4}{y-2}
$$

- Multiply the numerators and denominators:
$$
\frac{y \cdot y^4}{(y+3)(y-2)} = \frac{y^5}{(y+3)(y-2)}
$$

- The expression is already simplified.

- Final Answer:
$$
\boxed{\frac{y^5}{(y+3)(y-2)}}
$$

---

#### Example 4:
$$
\frac{x-2}{x+3} \times \frac{x+1}{x-2}
$$

- Multiply the numerators and denominators:
$$
\frac{(x-2)(x+1)}{(x+3)(x-2)}
$$

- Cancel the common factor \( x-2 \) in the numerator and denominator (assuming \( x \neq 2 \)):
$$
\frac{(x-2)(x+1)}{(x+3)(x-2)} = \frac{x+1}{x+3}
$$

- Final Answer:
$$
\boxed{\frac{x+1}{x+3}}
$$

---

#### Example 5:
$$
\frac{(x+4)(x+2)}{x+7} \times \frac{x+7}{(x-2)(x-3)}
$$

- Multiply the numerators and denominators:
$$
\frac{(x+4)(x+2)(x+7)}{(x+7)(x-2)(x-3)}
$$

- Cancel the common factor \( x+7 \) in the numerator and denominator (assuming \( x \neq -7 \)):
$$
\frac{(x+4)(x+2)(x+7)}{(x+7)(x-2)(x-3)} = \frac{(x+4)(x+2)}{(x-2)(x-3)}
$$

- The expression is now simplified.

- Final Answer:
$$
\boxed{\frac{(x+4)(x+2)}{(x-2)(x-3)}}
$$

---

#### Example 6:
$$
\frac{(p+1)(q-3)}{(p-2)^2} \times \frac{p-2}{(q-2)(p-3)}
$$

- Multiply the numerators and denominators:
$$
\frac{(p+1)(q-3)(p-2)}{(p-2)^2(q-2)(p-3)}
$$

- Cancel the common factor \( p-2 \) in the numerator and denominator (assuming \( p \neq 2 \)):
$$
\frac{(p+1)(q-3)(p-2)}{(p-2)^2(q-2)(p-3)} = \frac{(p+1)(q-3)}{(p-2)(q-2)(p-3)}
$$

- The expression is now simplified.

- Final Answer:
$$
\boxed{\frac{(p+1)(q-3)}{(p-2)(q-2)(p-3)}}
$$

---

#### Example 7:
$$
\frac{x^2 + 3x - 4}{x+2} \times \frac{x+2}{6(x+4)}
$$

- First, factor the quadratic expression \( x^2 + 3x - 4 \):
$$
x^2 + 3x - 4 = (x+4)(x-1)
$$

- Substitute the factored form into the expression:
$$
\frac{(x+4)(x-1)}{x+2} \times \frac{x+2}{6(x+4)}
$$

- Multiply the numerators and denominators:
$$
\frac{(x+4)(x-1)(x+2)}{(x+2) \cdot 6(x+4)}
$$

- Cancel the common factors \( x+2 \) and \( x+4 \) in the numerator and denominator (assuming \( x \neq -2 \) and \( x \neq -4 \)):
$$
\frac{(x+4)(x-1)(x+2)}{(x+2) \cdot 6(x+4)} = \frac{x-1}{6}
$$

- Final Answer:
$$
\boxed{\frac{x-1}{6}}
$$

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#### Example 8:
$$
\frac{x^2 + 4x - 5}{x^2 - 2x - 3} \times \frac{x^2 + 6x + 5}{x^2 - 4x + 3}
$$

- Factor each quadratic expression:
1. \( x^2 + 4x - 5 = (x+5)(x-1) \)
2. \( x^2 - 2x - 3 = (x-3)(x+1) \)
3. \( x^2 + 6x + 5 = (x+5)(x+1) \)
4. \( x^2 - 4x + 3 = (x-3)(x-1) \)

- Substitute the factored forms into the expression:
$$
\frac{(x+5)(x-1)}{(x-3)(x+1)} \times \frac{(x+5)(x+1)}{(x-3)(x-1)}
$$

- Multiply the numerators and denominators:
$$
\frac{(x+5)(x-1)(x+5)(x+1)}{(x-3)(x+1)(x-3)(x-1)}
$$

- Cancel the common factors \( x+1 \) and \( x-1 \) in the numerator and denominator (assuming \( x \neq -1 \) and \( x \neq 1 \)):
$$
\frac{(x+5)(x-1)(x+5)(x+1)}{(x-3)(x+1)(x-3)(x-1)} = \frac{(x+5)^2}{(x-3)^2}
$$

- Final Answer:
$$
\boxed{\frac{(x+5)^2}{(x-3)^2}}
$$

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Summary of Final Answers:

1. \( \boxed{\frac{x^2}{3}} \)
2. \( \boxed{\frac{am}{10b}} \)
3. \( \boxed{\frac{y^5}{(y+3)(y-2)}} \)
4. \( \boxed{\frac{x+1}{x+3}} \)
5. \( \boxed{\frac{(x+4)(x+2)}{(x-2)(x-3)}} \)
6. \( \boxed{\frac{(p+1)(q-3)}{(p-2)(q-2)(p-3)}} \)
7. \( \boxed{\frac{x-1}{6}} \)
8. \( \boxed{\frac{(x+5)^2}{(x-3)^2}} \)