To divide the polynomial $4x^3 - 4x^2 - 6$ by $x - 3$, we can use polynomial long division.

Here are the steps:

1. Divide the leading term of the dividend (the first term of $4x^3 - 4x^2 - 6$) by the leading term of the divisor $(x - 3)$.

   $\frac{4x^3}{x} = 4x^2$

   Now, multiply $4x^2$ by the divisor $(x - 3)$:

   $4x^2 \cdot (x - 3) = 4x^3 - 12x^2$

2. Subtract this result from the original polynomial:

   $(4x^3 - 4x^2 - 6) - (4x^3 - 12x^2) = (4x^3 - 4x^2 - 6) - 4x^3 + 12x^2 = 8x^2 - 6$

3. Repeat the process with the new dividend $8x^2 - 6$. Divide the leading term $8x^2$ by $x$:

   $\frac{8x^2}{x} = 8x$

   Now, multiply $8x$ by the divisor:

   $8x \cdot (x - 3) = 8x^2 - 24x$

4. Subtract this from the current dividend:

   $(8x^2 - 6) - (8x^2 - 24x) = 8x^2 - 6 - 8x^2 + 24x = 24x - 6$

5. Repeat once more, divide $24x$ by $x$:

   $\frac{24x}{x} = 24$

   Now, multiply $24$ by the divisor:

   $24 \cdot (x - 3) = 24x - 72$

6. Subtract:

   $(24x - 6) - (24x - 72) = 24x - 6 - 24x + 72 = 66$

So, the result of the division is:

$4x^2 + 8x + 24 + \frac{66}{x - 3}$

Final Answer:

$4x^2 + 8x + 24 + \frac{66}{x - 3}$

Would you like more details on polynomial long division, or do you have any questions?

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Here are five related questions to practice:

1. Divide $6x^3 - 5x^2 + 4$ by $x - 2$.
2. Perform the division: $x^4 + 3x^3 - 2x$ by $x + 1$.
3. Divide $5x^2 - 7x + 9$ by $x + 2$.
4. Divide $8x^3 - 2x + 5$ by $2x - 1$.
5. Simplify the expression $2x^4 - 4x^3 + x - 5$ divided by $x - 2$.

Tip:

Always check for factoring opportunities before using long division; it can simplify the process!