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)

To find \(\left( f - g + h \right)(x)\), we start by expressing the given functions in a format that allows us to perform the operations:

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

We aim to find:
\[
\left( f - g + h \right)(x) = f(x) - g(x) + h(x)
\]

### Step 1: Simplify Each Function

1. **Simplify \( f(x) = \frac{3}{x^2 - 4} \)**

   Notice that \( x^2 - 4 \) is a difference of squares:
   \[
   x^2 - 4 = (x - 2)(x + 2)
   \]
   So:
   \[
   f(x) = \frac{3}{(x - 2)(x + 2)}
   \]

2. **\( g(x) = \frac{10}{x^4 - x^2 + 16} \)**

   In this case, \( x^4 - x^2 + 16 \) does not factor easily into simpler terms, so we will leave \( g(x) \) as it is for now.

3. **Simplify \( h(x) = \frac{5}{x^2 + 4x + 4} \)**

   Notice that \( x^2 + 4x + 4 \) is a perfect square:
   \[
   x^2 + 4x + 4 = (x + 2)^2
   \]
   So:
   \[
   h(x) = \frac{5}{(x + 2)^2}
   \]

### Step 2: Combine the Functions

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

Combining these:

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

### Final Answer

The expression for \(\left( f - g + h \right)(x)\) is:

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

This is the simplified form of the given expression.

### Do You Want More Details?

Would you like me to further simplify this expression or perform any specific operation, like finding a common denominator? Let me know!

### Related Questions:

1. How can you find the common denominator for \(\frac{3}{(x-2)(x+2)}\) and \(\frac{5}{(x+2)^2}\)?
2. How would you factor \(x^4 - x^2 + 16\) to further simplify \(g(x)\)?
3. What are the domain restrictions for the function \(\left(f - g + h\right)(x)\)?
4. How do you find the zeros of \(\left(f - g + h\right)(x)\)?
5. What is the graphical representation of \(\left(f - g + h\right)(x)\)?

### Tip:

When simplifying expressions with multiple fractions, always check if any terms can be factored or simplified first. This can make finding a common denominator much easier!

Learn how to simplify and combine the fractional functions f(x), g(x), and h(x) in this math problem. Understand the steps involved in expressing each function in its simplest form and then combining them into a single expression. Suitable for high school students studying advanced algebra.

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