To divide \( 8x^3 + 30x^2 + 30x + 9 \) by \( 4x + 3 \) using long division, follow these steps:
Step 1: Set up the long division
___________________________
4x + 3 | 8x^3 + 30x^2 + 30x + 9
Step 2: Divide the leading term
Divide the leading term of the dividend (\(8x^3\)) by the leading term of the divisor (\(4x\)): \[ \frac{8x^3}{4x} = 2x^2 \]
Step 3: Multiply and subtract
Now, multiply \(2x^2\) by the entire divisor \(4x + 3\) and subtract from the original polynomial:
\[ 2x^2(4x + 3) = 8x^3 + 6x^2 \]
Subtract this from the original polynomial:
\[ \begin{array}{r} 8x^3 + 30x^2 + 30x + 9 \
- (8x^3 + 6x^2) \ \hline 0 + 24x^2 + 30x + 9 \ \end{array} \]
Step 4: Repeat the process
Now, bring down the next term(s) and repeat. We will now divide \(24x^2\) by \(4x\):
\[ \frac{24x^2}{4x} = 6x \]
Multiply \(6x\) by \(4x + 3\):
\[ 6x(4x + 3) = 24x^2 + 18x \]
Subtract this from the current polynomial:
\[ \begin{array}{r} 24x^2 + 30x + 9 \
- (24x^2 + 18x) \ \hline 0 + 12x + 9 \ \end{array} \]
Step 5: Repeat again
Now, divide \(12x\) by \(4x\):
\[ \frac{12x}{4x} = 3 \]
Multiply \(3\) by \(4x + 3\):
\[ 3(4x + 3) = 12x + 9 \]
Subtract this from the current polynomial:
\[ \begin{array}{r} 12x + 9 \
- (12x + 9) \ \hline 0 \ \end{array} \]
Final Result
Since there is no remainder, we completed the division.
The answer to \( \frac{8x^3 + 30x^2 + 30x + 9}{4x + 3} \) is:
\[ \text{Answer: } 2x^2 + 6x + 3 \]