Use the long division method to find the result when 9, x, cubed, plus, 30, x, squared, plus, 30, x, plus, 89*x*3+30*x*2+30*x*+8 is divided by 3, x, plus, 43*x*+4.

Let's use the long division method to divide \( 9x^3 + 30x^2 + 30x + 8 \) by \( 3x + 4 \).

### Step 1: Set up the division
We start by dividing the leading term of the dividend by the leading term of the divisor.

- Dividend: \( 9x^3 + 30x^2 + 30x + 8 \)
- Divisor: \( 3x + 4 \)

First, divide the leading term of the dividend, \( 9x^3 \), by the leading term of the divisor, \( 3x \):

\[
\frac{9x^3}{3x} = 3x^2
\]

### Step 2: Multiply the entire divisor by this result
Now, multiply \( 3x^2 \) by \( 3x + 4 \):

\[
3x^2 \cdot (3x + 4) = 9x^3 + 12x^2
\]

### Step 3: Subtract this result from the original polynomial
Subtract \( 9x^3 + 12x^2 \) from the original polynomial:

\[
(9x^3 + 30x^2 + 30x + 8) - (9x^3 + 12x^2) = 18x^2 + 30x + 8
\]

### Step 4: Repeat the process
Next, divide the leading term of the new polynomial \( 18x^2 \) by the leading term of the divisor \( 3x \):

\[
\frac{18x^2}{3x} = 6x
\]

Multiply \( 6x \) by the divisor \( 3x + 4 \):

\[
6x \cdot (3x + 4) = 18x^2 + 24x
\]

Subtract this from the current polynomial:

\[
(18x^2 + 30x + 8) - (18x^2 + 24x) = 6x + 8
\]

### Step 5: Continue the process
Now, divide \( 6x \) by \( 3x \):

\[
\frac{6x}{3x} = 2
\]

Multiply \( 2 \) by \( 3x + 4 \):

\[
2 \cdot (3x + 4) = 6x + 8
\]

Subtract this from the current polynomial:

\[
(6x + 8) - (6x + 8) = 0
\]

### Final Result
The remainder is \( 0 \), and the quotient is \( 3x^2 + 6x + 2 \).

So, the result of the division is:

\[
\boxed{3x^2 + 6x + 2}
\]

Would you like more details on any part of this process, or do you have any questions?

Here are some related questions you might find helpful:
1. How do you identify if a polynomial can be exactly divided by another polynomial?
2. How does the remainder affect the final result in polynomial division?
3. What are the practical applications of polynomial division in real-world problems?
4. How does synthetic division handle similar problems more efficiently?
5. Can polynomial division be extended to complex numbers or higher dimensions?

**Tip:** Always double-check your subtraction steps in polynomial division to ensure accuracy.

Learn how to use the long division method to divide 9x^3 + 30x^2 + 30x + 8 by 3x + 4 with detailed step-by-step instructions and a clear explanation of the process. Suitable for high school students studying polynomial division.

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