We are given the function:

$f(x) = -x - 2$

The task is to find the simplified expression for:

$f(x) - 3f(x) - 3$

Step 1: Substitute $f(x)$ in the expression.

We know that $f(x) = -x - 2$. So, substitute this into the expression:

$f(x) - 3f(x) - 3 = (-x - 2) - 3(-x - 2) - 3$

Step 2: Simplify the expression.

Distribute the $-3$ to the terms inside the parentheses:

$= (-x - 2) - 3(-x) - 3(-2) - 3$ $= (-x - 2) + 3x + 6 - 3$

Step 3: Combine like terms.

$= -x + 3x - 2 + 6 - 3$

Simplify:

$= 2x + 1$

Final Answer:

The simplified polynomial is:

$2x + 1$

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore:

1. How do you find the value of $f(x)$ at a specific point?
2. What happens if you modify $f(x)$ by multiplying it by a constant?
3. How do you simplify a polynomial involving multiple terms?
4. What is the effect of distributing negative signs in expressions?
5. How do you solve equations involving functions and constants?

Tip: Always distribute carefully when working with negative coefficients to avoid sign errors.