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The image shows a polynomial long division problem where you are asked to divide \( 18x^4 + 12x^3 + 3x^2 \) by \( 3x^2 + 2 \) and find the quotient \( q(x) \) and remainder \( r(x) \). Let's go through the steps for polynomial division.

### Step 1: Divide the first terms
Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{18x^4}{3x^2} = 6x^2
\]
This is the first term of the quotient.

### Step 2: Multiply and subtract
Now, multiply \( 6x^2 \) by the divisor \( 3x^2 + 2 \):
\[
6x^2(3x^2 + 2) = 18x^4 + 12x^2
\]
Subtract this from the original polynomial:
\[
(18x^4 + 12x^3 + 3x^2) - (18x^4 + 12x^2) = 12x^3 - 9x^2
\]

### Step 3: Divide the next term
Now, divide the leading term of the new polynomial \( 12x^3 \) by the leading term of the divisor \( 3x^2 \):
\[
\frac{12x^3}{3x^2} = 4x
\]
This is the next term of the quotient.

### Step 4: Multiply and subtract
Multiply \( 4x \) by the divisor \( 3x^2 + 2 \):
\[
4x(3x^2 + 2) = 12x^3 + 8x
\]
Subtract this from \( 12x^3 - 9x^2 \):
\[
(12x^3 - 9x^2) - (12x^3 + 8x) = -9x^2 - 8x
\]

### Step 5: Divide the next term
Now, divide \( -9x^2 \) by \( 3x^2 \):
\[
\frac{-9x^2}{3x^2} = -3
\]
This is the last term of the quotient.

### Step 6: Multiply and subtract
Multiply \( -3 \) by the divisor \( 3x^2 + 2 \):
\[
-3(3x^2 + 2) = -9x^2 - 6
\]
Subtract this from \( -9x^2 - 8x \):
\[
(-9x^2 - 8x) - (-9x^2 - 6) = -8x + 6
\]
This is the remainder.

### Final Answer:
The quotient \( q(x) \) is:
\[
6x^2 + 4x - 3
\]
The remainder \( r(x) \) is:
\[
-8x + 6
\]

Would you like further details or explanations? 

Here are some related questions:
1. What is the process of synthetic division, and how does it differ from polynomial long division?
2. How can you verify the result of a polynomial division?
3. What are the applications of the remainder theorem in algebra?
4. What

Divide 18x^4 + 12x^3 + 3x^2 by 3x^2 + 2 using polynomial long division. State the quotient q(x) and the remainder r(x).

Learn how to divide the polynomial 18x^4 + 12x^3 + 3x^2 by 3x^2 + 2 using polynomial long division. This problem covers essential algebraic concepts like dividing polynomials, finding quotients, and determining remainders, following step-by-step calculations. Suitable for students in Grades 9-12.

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