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I can see the question asks to perform polynomial operations with two given functions, \( f(x) = -8x^4 + x^2 \) and \( g(x) = 4x^4 + 2x - 1 \), and there are three different operations to evaluate:

1. \( (f + g)(x) \)
2. \( (f - g)(x) \)
3. \( (f \cdot g)(x) \)

Let's solve each of these step by step.

### 1. \( (f + g)(x) \)
To find \( (f + g)(x) \), we simply add the two functions:
\[
f(x) + g(x) = (-8x^4 + x^2) + (4x^4 + 2x - 1)
\]
Now, combine like terms:
\[
= (-8x^4 + 4x^4) + x^2 + 2x - 1
\]
\[
= -4x^4 + x^2 + 2x - 1
\]
So, \( (f + g)(x) = -4x^4 + x^2 + 2x - 1 \).

### 2. \( (f - g)(x) \)
For \( (f - g)(x) \), subtract the two functions:
\[
f(x) - g(x) = (-8x^4 + x^2) - (4x^4 + 2x - 1)
\]
Distribute the negative sign:
\[
= (-8x^4 + x^2) - 4x^4 - 2x + 1
\]
Now, combine like terms:
\[
= (-8x^4 - 4x^4) + x^2 - 2x + 1
\]
\[
= -12x^4 + x^2 - 2x + 1
\]
So, \( (f - g)(x) = -12x^4 + x^2 - 2x + 1 \).

### 3. \( (f \cdot g)(x) \)
Now for the multiplication \( (f \cdot g)(x) \):
\[
f(x) \cdot g(x) = (-8x^4 + x^2) \cdot (4x^4 + 2x - 1)
\]
We'll distribute each term in \( f(x) \) by each term in \( g(x) \):
\[
= (-8x^4)(4x^4) + (-8x^4)(2x) + (-8x^4)(-1) + (x^2)(4x^4) + (x^2)(2x) + (x^2)(-1)
\]
Now calculate each product:
\[
= -32x^8 - 16x^5 + 8x^4 + 4x^6 + 2x^3 - x^2
\]
The resulting expression in descending order of powers is:
\[
= -32x^8 + 4x^6 - 16x^5 + 8x^4 + 2x^3 - x^2
\]
So, \( (f \cdot g)(x) = -32x^8 + 4x^6 - 16x^5 + 8x^4 + 2x^3 - x^2 \).

Let me know if you'd like more details or if you have any questions! Here are some related questions for further exploration:

1. What happens if you add or subtract two polynomials with different degrees?
2. How would the expression change if one of the polynomials was a cubic function?
3. How do you apply the distributive property in polynomial multiplication?
4. Can polynomials of higher degree always be combined in operations like addition or subtraction?
5. How would you graph the sum and difference of these functions?

**Tip:** Always double-check the combination of like terms in polynomial addition and subtraction to avoid mistakes in the final expression.

Given the functions f(x) = -8x^4 + x^2 and g(x) = 4x^4 + 2x - 1, determine each of the following. Give your answers as simplified expressions written in descending order: (f + g)(x), (f - g)(x), (f * g)(x).

Explore polynomial operations involving the functions f(x) = -8x^4 + x^2 and g(x) = 4x^4 + 2x - 1. Learn how to evaluate (f + g)(x), (f - g)(x), and (f * g)(x) with step-by-step solutions. Suitable for students in Grades 9-12.

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