Let's go through the problems step-by-step.

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Problem g

Expression:
$\frac{2x^2 + x - 10}{x^2 + 2x - 8} + \frac{4x^2 + 20x + 25}{x + 4}$

1. Factor each part where possible:

   • For $2x^2 + x - 10$, factor as $(2x - 3)(x + 2)$.
   • For $x^2 + 2x - 8$, factor as $(x + 4)(x - 2)$.
   • For $4x^2 + 20x + 25$, factor as $(2x + 5)^2$.

2. Rewrite the fractions and find the common denominator.

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Problem h

Expression:
$\frac{12x - 8}{x^2 - 2x - 15} - \frac{x^2 - x - 12}{3x^2 - 9x - 12}$

1. Factor each term:

   • For $12x - 8$, factor out the 4: $4(3x - 2)$.
   • For $x^2 - 2x - 15$, factor as $(x - 5)(x + 3)$.
   • For $x^2 - x - 12$, factor as $(x - 4)(x + 3)$.
   • For $3x^2 - 9x - 12$, factor as $3(x - 4)(x + 3)$.

2. Rewrite and subtract the fractions using the common denominator.

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Problem i

Expression:
$\frac{\frac{x + 2}{4}}{\frac{3}{x + 2}}$

1. Simplify by multiplying by the reciprocal of the denominator: $\frac{x + 2}{4} \cdot \frac{x + 2}{3} = \frac{(x + 2)^2}{12}$

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Problem j

Expression:
$\frac{\frac{x + 4}{x^2}}{\frac{x - 5}{x - 5}}$

1. Since $\frac{x - 5}{x - 5} = 1$, simplify the expression to $\frac{x + 4}{x^2}$.

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Problem 5

Part (a)

$\frac{1}{3} + \frac{2}{5} = \frac{5 + 6}{15} = \frac{11}{15}$

Part (b)

$\frac{4}{9} - \frac{10}{11} = \frac{44 - 90}{99} = -\frac{46}{99}$

Part (c)

$\frac{1}{x} + \frac{4 + x}{x} = \frac{5 + x}{x}$

Part (d)

$\frac{4}{x} - \frac{1}{2x} = \frac{8 - 1}{2x} = \frac{7}{2x}$

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Would you like more detailed solutions for any specific problem?

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Follow-up Questions

1. How do you find the least common denominator for adding rational expressions?
2. What steps are necessary to simplify complex fractions?
3. Why is factoring important in simplifying rational expressions?
4. How does cross-multiplication work when comparing fractions?
5. What are some common mistakes to avoid when adding and subtracting fractions?

Tip

Always double-check for factors common to both the numerator and denominator to simplify the fraction fully.