In the given problem, you are tasked with determining $g(x + a) - g(x)$ for the function:

$g(x) = -2x - 2$

To solve this, let's break it down:

1. Substitute $x + a$ into the function:

   $g(x + a)$ means replacing $x$ in the given function with $x + a$: $g(x + a) = -2(x + a) - 2$ Simplifying this: $g(x + a) = -2x - 2a - 2$

2. Find $g(x + a) - g(x)$:

   Now, subtract $g(x) = -2x - 2$ from $g(x + a)$: $g(x + a) - g(x) = (-2x - 2a - 2) - (-2x - 2)$ Simplify the expression: $g(x + a) - g(x) = -2x - 2a - 2 + 2x + 2$ The terms $-2x$ and $+2x$ cancel out, leaving: $g(x + a) - g(x) = -2a$

Therefore, the expression $g(x + a) - g(x)$ simplifies to: $g(x + a) - g(x) = -2a$

Would you like a more detailed explanation or have further questions? Here are some related questions:

1. How does this approach apply to other linear functions?
2. Can you determine the expression for $g(x + a) - g(x)$ for a different function, such as $g(x) = 3x + 4$?
3. What happens if the function $g(x)$ is quadratic instead of linear?
4. How would this expression change if $g(x) = -2x^2 - 2$?
5. Could you solve for $a$ if $g(x + a) - g(x) = -4a$?

Tip: For any linear function, $g(x) = mx + b$, the expression $g(x + a) - g(x)$ will always simplify to $ma$, where $m$ is the coefficient of $x$.