To find $g(g(x))$, we need to substitute $g(x)$ into itself. The given function for $g(x)$ is:

$g(x) = x^2 + 2x - 1$

Now, let's substitute this expression into $g(x)$ again. So, we are finding:

$g(g(x)) = g(x^2 + 2x - 1)$

Since $g(x) = x^2 + 2x - 1$, we replace $x$ in this formula with $x^2 + 2x - 1$:

$g(g(x)) = (x^2 + 2x - 1)^2 + 2(x^2 + 2x - 1) - 1$

Let's simplify this step by step:

1. Square $(x^2 + 2x - 1)$:

$(x^2 + 2x - 1)^2 = (x^2 + 2x - 1)(x^2 + 2x - 1)$

Expanding this:

$= x^4 + 2x^3 - x^2 + 2x^3 + 4x^2 - 2x - x^2 - 2x + 1$

Combining like terms:

$= x^4 + 4x^3 + 2x^2 - 4x + 1$

2. Multiply $2(x^2 + 2x - 1)$:

$2(x^2 + 2x - 1) = 2x^2 + 4x - 2$

3. Substitute and combine all terms:

$g(g(x)) = (x^4 + 4x^3 + 2x^2 - 4x + 1) + (2x^2 + 4x - 2) - 1$

Now, combine like terms:

$g(g(x)) = x^4 + 4x^3 + (2x^2 + 2x^2) + (-4x + 4x) + (1 - 2 - 1)$

Simplifying:

$g(g(x)) = x^4 + 4x^3 + 4x^2 + 0x - 2$

Thus, the final expression is:

$g(g(x)) = x^4 + 4x^3 + 4x^2 - 2$

Do you have any questions or would you like further details?

Here are 5 related questions for you to explore:

1. How would you find $f(f(x))$ given $f(x) = 5x + 16$?
2. What is the composition $f(g(x))$ for the same functions?
3. What is the derivative of $g(g(x))$?
4. What would be $g^{-1}(x)$, the inverse of $g(x)$?
5. How do you solve $g(g(x)) = 0$?

Tip: When working with composition of functions, it's essential to substitute carefully and combine like terms step by step to avoid mistakes.