To find \( g(x+1) - g(x-1) \) for the given function \( g(x) = x^2 - 4x \), we need to calculate \( g(x+1) \) and \( g(x-1) \) first, then find their difference.

First, calculate \( g(x+1) \):

\[
g(x+1) = (x+1)^2 - 4(x+1)
\]

Simplify:

\[
(x+1)^2 = x^2 + 2x + 1
\]
\[
4(x+1) = 4x + 4
\]
\[
g(x+1) = x^2 + 2x + 1 - 4x - 4 = x^2 - 2x - 3
\]

Next, calculate \( g(x-1) \):

\[
g(x-1) = (x-1)^2 - 4(x-1)
\]

Simplify:

\[
(x-1)^2 = x^2 - 2x + 1
\]
\[
4(x-1) = 4x - 4
\]
\[
g(x-1) = x^2 - 2x + 1 - 4x + 4 = x^2 - 6x + 5
\]

Now, find the difference \( g(x+1) - g(x-1) \):

\[
g(x+1) - g(x-1) = (x^2 - 2x - 3) - (x^2 - 6x + 5)
\]

Simplify the expression:

\[
g(x+1) - g(x-1) = x^2 - 2x - 3 - x^2 + 6x - 5
\]
\[
g(x+1) - g(x-1) = -2x + 6x - 3 - 5
\]
\[
g(x+1) - g(x-1) = 4x - 8
\]

So, the result is:

\[
g(x+1) - g(x-1) = 4x - 8
\]

Would you like further details or have any questions about this solution?

Here are some related questions you might find helpful:
1. How would you find \( g(x+2) - g(x) \) for the same function \( g(x) \)?
2. What is the derivative of \( g(x) = x^2 - 4x \)?
3. How would you integrate \( g(x) = x^2 - 4x \)?
4. Can you find the roots of the function \( g(x) = x^2 - 4x \)?
5. How would you solve \( g(x) = 0 \) for \( g(x) = x^2 - 4x \)?
6. What is the vertex form of the quadratic function \( g(x) = x^2 - 4x \)?
7. How does the function \( g(x) = x^2 - 4x \) transform if we add a constant \( c \)?
8. What is the maximum or minimum value of the function \( g(x) = x^2 - 4x \)?

**Tip:** When simplifying expressions, carefully combine like terms and keep track of the signs to avoid mistakes.

Learn how to calculate g(x+1) - g(x-1) for the quadratic function g(x) = x^2 - 4x. Step-by-step solution including simplification and explanation of function operations.

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