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The question in the image asks you to use long division to divide the polynomial \( 3x^4 + 13x^3 + 2x^2 + 10x - 18 \) by \( 3x^2 + x + 4 \). If there is a remainder, it should be expressed in the form \( q(x) + \frac{r(x)}{d(x)} \), where \( q(x) \) is the quotient, \( r(x) \) is the remainder, and \( d(x) \) is the divisor.

Let's proceed step by step using polynomial long division:

### Step 1: Divide the first term of the dividend by the first term of the divisor:
\[
\frac{3x^4}{3x^2} = x^2
\]
So, the first term of the quotient \( q(x) \) is \( x^2 \).

### Step 2: Multiply the entire divisor by \( x^2 \):
\[
x^2 \cdot (3x^2 + x + 4) = 3x^4 + x^3 + 4x^2
\]

### Step 3: Subtract this from the original polynomial:
\[
(3x^4 + 13x^3 + 2x^2 + 10x - 18) - (3x^4 + x^3 + 4x^2) = 12x^3 - 2x^2 + 10x - 18
\]

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

### Step 5: Multiply the entire divisor by \( 4x \):
\[
4x \cdot (3x^2 + x + 4) = 12x^3 + 4x^2 + 16x
\]

### Step 6: Subtract this from the current polynomial:
\[
(12x^3 - 2x^2 + 10x - 18) - (12x^3 + 4x^2 + 16x) = -6x^2 - 6x - 18
\]

### Step 7: Divide the first term of the new polynomial by the first term of the divisor:
\[
\frac{-6x^2}{3x^2} = -2
\]
So, the next term in the quotient is \( -2 \).

### Step 8: Multiply the entire divisor by \( -2 \):
\[
-2 \cdot (3x^2 + x + 4) = -6x^2 - 2x - 8
\]

### Step 9: Subtract this from the current polynomial:
\[
(-6x^2 - 6x - 18) - (-6x^2 - 2x - 8) = -4x - 10
\]

Now, the degree of the remainder, \( -4x - 10 \), is less than the degree of the divisor \( 3x^2 + x + 4 \), so we have reached the remainder.

### Final Answer:
The quotient is:
\[
q(x) = x^2 + 4x - 2
\]
The remainder is:
\[
r(x) = -4x - 10
\]
Thus, the final answer in the required form is:
\[
x^2 + 4x - 2 + \frac{-4x - 10}{3x^2 + x + 4}
\]

Would you like any more details on this solution? 

Here are some related questions for further practice:
1. Can you perform polynomial long division for \( 4x^3 + 2x^2 - x + 7 \) divided by \( x + 1 \)?
2. What happens if the degree of the divisor is greater than the degree of the dividend?
3. How does synthetic division differ from polynomial long division?
4. What is the remainder theorem, and how does it relate to polynomial division?
5. Can you verify the solution by multiplying the quotient and divisor and adding the remainder?

**Tip:** When dividing polynomials, always align the terms according to their degrees to avoid errors in the subtraction steps.

Use the long division method to find the result when 3x^4 + 13x^3 + 2x^2 + 10x - 18 is divided by 3x^2 + x + 4. If there is a remainder, express the result in the form q(x) + r(x)/d(x).

Step-by-step guide on using the long division method to divide polynomial 3x^4 + 13x^3 + 2x^2 + 10x - 18 by 3x^2 + x + 4, including how to handle remainders in polynomial division. Suitable for high school students learning advanced algebra.

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