Let's use the long division method to divide \(9x^3 + 30x^2 + 30x + 12\) by \(3x + 4\).
Step 1: Set up the long division
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3x + 4 | 9x^3 + 30x^2 + 30x + 12
Step 2: Divide the leading terms
Divide \(9x^3\) by \(3x\):
\[ \frac{9x^3}{3x} = 3x^2 \]
Step 3: Multiply and subtract
Multiply \(3x^2\) by \(3x + 4\):
\[ 3x^2(3x + 4) = 9x^3 + 12x^2 \]
Now subtract:
\[ (9x^3 + 30x^2 + 30x + 12) - (9x^3 + 12x^2) = 18x^2 + 30x + 12 \]
Now our division looks like this:
3x^2
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3x + 4 | 9x^3 + 30x^2 + 30x + 12
- (9x^3 + 12x^2)
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18x^2 + 30x + 12
Step 4: Repeat the process
Now divide the leading term \(18x^2\) by \(3x\):
\[ \frac{18x^2}{3x} = 6x \]
Multiply \(6x\) by \(3x + 4\):
\[ 6x(3x + 4) = 18x^2 + 24x \]
Now subtract:
\[ (18x^2 + 30x + 12) - (18x^2 + 24x) = 6x + 12 \]
Now our long division looks like this:
3x^2 + 6x
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3x + 4 | 9x^3 + 30x^2 + 30x + 12
- (9x^3 + 12x^2)
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18x^2 + 30x + 12
- (18x^2 + 24x)
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6x + 12
Step 5: Continue the process
Now divide the leading term \(6x\) by \(3x\):
\[ \frac{6x}{3x} = 2 \]
Multiply \(2\) by \(3x + 4\):
\[ 2(3x + 4) = 6x + 8 \]
Now subtract:
\[ (6x + 12) - (6x + 8) = 4 \]
Final result of the long division
Now our division shows:
3x^2 + 6x + 2
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3x + 4 | 9x^3 + 30x^2 + 30x + 12
- (9x^3 + 12x^2)
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18x^2 + 30x + 12
- (18x^2 + 24x)
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6x + 12
- (6x + 8)
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4
Step 6: Write in the required form
The quotient \(q(x)\) is \(3x^2 + 6x + 2\), and the remainder \(r(x)\) is \(4\).
Thus, the result can be expressed as:
\[ 3x^2 + 6x + 2 + \frac{4}{3x + 4} \]
Final answer:
\[ \boxed{3x^2 + 6x + 2 + \frac{4}{3x + 4}} \]