We are given the expression:

$\frac{1}{x+4} + \frac{x}{x-4} + \frac{24 - x^2}{x^2 - 16}$

Step 1: Factor the denominator where possible

The denominator of the third fraction can be factored:

$x^2 - 16 = (x+4)(x-4)$

So we rewrite the third fraction:

$\frac{24 - x^2}{(x+4)(x-4)}$

Since $24 - x^2 = -(x^2 - 24)$, we rewrite it as:

$\frac{-(x^2 - 24)}{(x+4)(x-4)}$

Step 2: Find a common denominator

The least common denominator (LCD) for all fractions is:

$(x+4)(x-4)$

Rewriting each fraction with this common denominator:

1. $\frac{1}{x+4} = \frac{x-4}{(x+4)(x-4)}$
2. $\frac{x}{x-4} = \frac{x(x+4)}{(x+4)(x-4)}$
3. $\frac{24 - x^2}{(x+4)(x-4)} = \frac{-(x^2 - 24)}{(x+4)(x-4)}$

Step 3: Combine the numerators

$\frac{x-4}{(x+4)(x-4)} + \frac{x(x+4)}{(x+4)(x-4)} + \frac{-(x^2 - 24)}{(x+4)(x-4)}$

Now, simplifying the numerators:

$(x - 4) + x(x + 4) - (x^2 - 24)$

Expanding terms:

$x - 4 + x^2 + 4x - x^2 + 24$

$x - 4 + 4x + 24$

$5x + 20$

Factoring:

$5(x + 4)$

Step 4: Simplify the fraction

$\frac{5(x+4)}{(x+4)(x-4)}$

Cancel $x+4$ (as long as $x \neq -4$):

$\frac{5}{x-4}$

Final Answer:

$\frac{5}{x-4}, \quad x \neq \pm 4$

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Related Questions:

1. How do you find the least common denominator (LCD) in rational expressions?
2. What are the restrictions on the variable $x$ in rational expressions?
3. How do you simplify complex fractions step by step?
4. Why does factoring help in simplifying algebraic fractions?
5. What happens if you substitute $x = 4$ or $x = -4$ into the original expression?

Tip:

Always check for restrictions in the domain before simplifying rational expressions to avoid dividing by zero!