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To find \((g - f)(x)\), we need to subtract \(f(x)\) from \(g(x)\).

Given:
\[
f(x) = 4x^3 + 3x^2 - 5x + 20
\]
\[
g(x) = 9x^3 - 4x^2 + 10x - 55
\]

### Step 1: Subtract \(f(x)\) from \(g(x)\)
\[
(g - f)(x) = g(x) - f(x)
\]
Substitute \(f(x)\) and \(g(x)\):
\[
(g - f)(x) = (9x^3 - 4x^2 + 10x - 55) - (4x^3 + 3x^2 - 5x + 20)
\]

### Step 2: Distribute the Negative Sign
\[
= 9x^3 - 4x^2 + 10x - 55 - 4x^3 - 3x^2 + 5x - 20
\]

### Step 3: Combine Like Terms
- For \(x^3\): \(9x^3 - 4x^3 = 5x^3\)
- For \(x^2\): \(-4x^2 - 3x^2 = -7x^2\)
- For \(x\): \(10x + 5x = 15x\)
- Constant terms: \(-55 - 20 = -75\)

Putting it all together:
\[
(g - f)(x) = 5x^3 - 7x^2 + 15x - 75
\]

### Answer
The correct choice is:
**(D) \(5x^3 - 7x^2 + 15x - 75\)**

Would you like further details on any part of this process?

### Related Questions
1. What is the result of adding \(f(x)\) and \(g(x)\) instead of subtracting?
2. How would \((g - f)(x)\) change if \(f(x)\) had a different constant term?
3. How do you find the derivative of \((g - f)(x)\)?
4. How would the answer change if the powers of \(x\) were different in \(f(x)\) and \(g(x)\)?
5. How does the subtraction of two polynomials relate to their graphs?

### Tip
When subtracting polynomials, carefully distribute the negative sign across each term in the second polynomial to avoid errors.

If f(x) = 4x^3 + 3x^2 - 5x + 20 and g(x) = 9x^3 - 4x^2 + 10x - 55, what is (g - f)(x)?

Learn how to find the difference between two polynomial functions, f(x) and g(x), with a step-by-step solution. This example problem focuses on subtracting polynomials and combining like terms, suitable for students in Grades 10-12.

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