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The problem asks to divide \( x^3 - 4x^2 - 4x + 21 \) by \( x - 3 \) using polynomial long division. I will guide you through the division process step-by-step:

### Step 1: Setup
You need to divide \( x^3 - 4x^2 - 4x + 21 \) by \( x - 3 \).

### Step 2: First Division
Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \). 

\[
\frac{x^3}{x} = x^2
\]

So, the first term of the quotient is \( x^2 \).

### Step 3: Multiply and Subtract
Now multiply \( x^2 \) by \( x - 3 \):

\[
x^2 \cdot (x - 3) = x^3 - 3x^2
\]

Subtract this from the original polynomial:

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

### Step 4: Second Division
Now divide \( -x^2 \) by \( x \):

\[
\frac{-x^2}{x} = -x
\]

So, the next term in the quotient is \( -x \).

### Step 5: Multiply and Subtract
Multiply \( -x \) by \( x - 3 \):

\[
-x \cdot (x - 3) = -x^2 + 3x
\]

Subtract this from the current polynomial:

\[
(-x^2 - 4x) - (-x^2 + 3x) = -7x
\]

### Step 6: Third Division
Now divide \( -7x \) by \( x \):

\[
\frac{-7x}{x} = -7
\]

So, the next term in the quotient is \( -7 \).

### Step 7: Multiply and Subtract
Multiply \( -7 \) by \( x - 3 \):

\[
-7 \cdot (x - 3) = -7x + 21
\]

Subtract this from the remaining polynomial:

\[
(-7x + 21) - (-7x + 21) = 0
\]

### Final Answer:
The quotient is \( x^2 - x - 7 \).

Would you like to review more details or ask any questions?

Here are five relative questions to expand your understanding:

1. How is synthetic division different from long division of polynomials?
2. What is the remainder theorem and how does it apply to this problem?
3. How can you check if your quotient is correct after division?
4. How does the degree of the polynomial affect the number of terms in the quotient?
5. What happens if there is a remainder in polynomial division?

**Tip:** When dividing polynomials, always make sure you align terms by their degrees during the subtraction process to avoid mistakes.

Fill in the missing values to find the quotient when x^3 - 4x^2 - 4x + 21 is divided by x - 3.

Learn how to solve the polynomial division problem: dividing x^3 - 4x^2 - 4x + 21 by x - 3. This step-by-step solution covers key algebraic concepts like polynomial long division. Suitable for students in Grades 9-11.

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