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You've provided an equation where the function notation \( g(g(x)) \) is expressed in two forms: \( g(x) = x^2 + 4x - 1 \) and \( g(g(x)) = g(x) - 2 \). We'll find \( g(g(x)) \) explicitly by substituting \( g(x) \) into itself and compare it to \( g(x) - 2 \) to verify the expressions. Let's solve this step-by-step.

1. Find \( g(g(x)) \):
   - First, we substitute \( g(x) \) into itself: \( g(g(x)) = g(x^2 + 4x - 1) \).
   - Then, substitute \( x \) in \( g(x) \) with \( x^2 + 4x - 1 \): \( g(x^2 + 4x - 1) = (x^2 + 4x - 1)^2 + 4(x^2 + 4x - 1) - 1 \).

2. Expand and simplify \( g(g(x)) \) to compare with \( g(x) - 2 \).The expression for \( g(g(x)) \) when expanded and simplified is \( x^4 + 8x^3 + 18x^2 + 8x - 4 \).

3. Verify if \( g(g(x)) \) matches \( g(x) - 2 \):
   - Calculate \( g(x) - 2 \): \( (x^2 + 4x - 1) - 2 \).
   - Simplify the result to compare with \( g(g(x)) \).It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like to attempt a different type of question, or can I help with a theoretical explanation instead?

Meanwhile, here are some related topics you might find interesting to explore:
1. How to solve quadratic equations and find their roots.
2. The concept of function composition in mathematics.
3. How to expand and simplify polynomial expressions.
4. Methods to verify if two algebraic expressions are equivalent.
5. The use of substitution in functions to derive new expressions.

**Tip:** When working with nested functions like \( g(g(x)) \), always ensure that each substitution step is clearly defined and simplified for ease of comparison and further calculations.

g(g(x)) = x^2 + 4x - 1 and g(g(x)) = g(x) - 2. Find g(g(x)) explicitly by substituting g(x) into itself and compare it to g(x) - 2.

Explore the step-by-step solution of the function composition problem g(g(x)) = g(x) - 2 with g(x) = x^2 + 4x - 1. Learn about key algebraic concepts such as substitution, polynomial expansion, and function simplification, suitable for high school students in Grades 10-12.

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