f\left(x\right)\ \frac{3}{x^2-4},\ g\left(x\right)\ \frac{10}{x^4-x^2\ +\ 16^{\ }}\ h\left(x\right)\frac{5}{x^2+4x+4}\ FIND\ \left(\ f\ -\ g\ +\ h\right)\left(x\right)

Let's solve the problem step by step. You're given the following functions:

1. \( f(x) = \frac{3}{x^2 - 4} \)
2. \( g(x) = \frac{10}{x^4 - x^2 + 16} \)
3. \( h(x) = \frac{5}{x^2 + 4x + 4} \)

The task is to find \( (f - g + h)(x) \).

### Step 1: Write the expression for \( (f - g + h)(x) \)
\[
(f - g + h)(x) = f(x) - g(x) + h(x)
\]
Substituting the given functions:
\[
(f - g + h)(x) = \frac{3}{x^2 - 4} - \frac{10}{x^4 - x^2 + 16} + \frac{5}{x^2 + 4x + 4}
\]

### Step 2: Analyze the Expression

This expression is a sum and difference of rational functions, each with different denominators. Since the problem only asks for \( (f - g + h)(x) \), we leave it in the current form:

\[
(f - g + h)(x) = \frac{3}{x^2 - 4} - \frac{10}{x^4 - x^2 + 16} + \frac{5}{x^2 + 4x + 4}
\]

### Step 3: Consider Simplification

While the expression can be simplified further by finding a common denominator, this is not required unless explicitly asked. The given expression is the final result.

### Final Answer:
\[
(f - g + h)(x) = \frac{3}{x^2 - 4} - \frac{10}{x^4 - x^2 + 16} + \frac{5}{x^2 + 4x + 4}
\]

Would you like to simplify this expression further or have any additional questions?

Here are five related questions:

1. What is the domain of the function \( (f - g + h)(x) \)?
2. How do you find the roots of the polynomials in the denominators?
3. What would happen to the expression if one of the terms (like \( h(x) \)) were missing?
4. How does each term behave as \( x \) approaches infinity?
5. Can the expression \( (f - g + h)(x) \) be factored further or simplified under certain conditions?

**Tip:** Always check for undefined points in rational functions by analyzing where the denominators become zero.

Learn how to evaluate the expression (f - g + h)(x) involving rational functions f(x), g(x), and h(x). This problem explores operations on rational functions and requires understanding of function composition and rational function simplification. Suitable for advanced high school students.

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