Let's simplify the given expressions step by step:

Part (a):

$\text{Simplify: } \frac{a^2 - b^2}{a^2 + b^2} \times \frac{a^2 + ab}{b^2 - a^2} \div a$

Step 1: Simplify each fraction individually.

• The first fraction is $\frac{a^2 - b^2}{a^2 + b^2}$. Notice that $a^2 - b^2$ is a difference of squares, which factors as: $\frac{(a - b)(a + b)}{a^2 + b^2}$ Since $a^2 + b^2$ does not factor nicely, we'll keep it as is.

• The second fraction is $\frac{a^2 + ab}{b^2 - a^2}$. The denominator $b^2 - a^2$ can also be factored using the difference of squares: $\frac{a^2 + ab}{-(a^2 - b^2)} = \frac{a^2 + ab}{-(a - b)(a + b)}$

Step 2: Combine the expressions.

The expression now becomes: $\frac{(a - b)(a + b)}{a^2 + b^2} \times \frac{a^2 + ab}{-(a - b)(a + b)} \div a$

Notice that $(a - b)(a + b)$ cancels out from the numerator and denominator: $\frac{1}{a^2 + b^2} \times \frac{a^2 + ab}{-1} \div a$

This simplifies to: $-\frac{a^2 + ab}{a(a^2 + b^2)}$

Finally, simplify by dividing both terms in the numerator by $a$: $-\frac{a + b}{a^2 + b^2}$

Part (b):

$\text{Simplify: } \frac{x^3 + 4x^2}{x^2 - 1} \times \frac{x^2 - 5x + 6}{x^2 - 3x} \div \frac{x^2 + 2x - 8}{x^2 - 1}$

Step 1: Factor each expression where possible.

• The first numerator $x^3 + 4x^2$ can be factored as: $x^2(x + 4)$ The first denominator $x^2 - 1$ is a difference of squares: $(x - 1)(x + 1)$

• The second numerator $x^2 - 5x + 6$ factors as: $(x - 2)(x - 3)$ The second denominator $x^2 - 3x$ can be factored by taking out the common factor: $x(x - 3)$

• The third fraction in the denominator $x^2 + 2x - 8$ factors as: $(x - 2)(x + 4)$ The third denominator $x^2 - 1$ is the same as the first denominator, which is: $(x - 1)(x + 1)$

Step 2: Combine the expressions.

The expression becomes: $\frac{x^2(x + 4)}{(x - 1)(x + 1)} \times \frac{(x - 2)(x - 3)}{x(x - 3)} \div \frac{(x - 2)(x + 4)}{(x - 1)(x + 1)}$

Step 3: Simplify by canceling out common factors.

• $x^2$ in the first term cancels with $x$ in the second term (leaving $x$ in the numerator).
• $(x - 3)$ cancels out.
• The third fraction can be inverted and multiplied:

The simplified expression now is: $\frac{x \times 1}{(x - 1)(x + 1)} \times \frac{(x - 1)(x + 1)}{(x - 2)(x + 4)}$

Since $(x - 1)(x + 1)$ cancels out: $\frac{x}{(x - 2)(x + 4)}$

Final Simplified Answers:

(a) $-\frac{a + b}{a^2 + b^2}$

(b) $\frac{x}{(x - 2)(x + 4)}$

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Do you want further details or have any questions about the steps?

Here are 5 related questions:

1. How do you factor a difference of squares?
2. What is the importance of factoring in simplifying algebraic expressions?
3. How do you multiply and divide algebraic fractions?
4. What are some common mistakes when simplifying complex expressions?
5. How do you identify common factors in algebraic expressions?

Tip: Always double-check that you fully cancel out common factors when simplifying expressions to avoid errors.