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To solve the given division problem, we will use polynomial long division. The expression to be divided is:

\[
(2x^4 + 15x^3 + 19x^2 + 4x - 11) \div (x + 6)
\]

### Step-by-step Polynomial Long Division:
1. **First Term**: Divide the first term of the dividend \(2x^4\) by the first term of the divisor \(x\). This gives:
   \[
   \frac{2x^4}{x} = 2x^3
   \]
   Now, multiply \(2x^3\) by the divisor \(x + 6\):
   \[
   2x^3(x + 6) = 2x^4 + 12x^3
   \]
   Subtract this from the original polynomial:
   \[
   (2x^4 + 15x^3 + 19x^2 + 4x - 11) - (2x^4 + 12x^3) = 3x^3 + 19x^2 + 4x - 11
   \]

2. **Second Term**: Divide the new first term \(3x^3\) by \(x\), giving:
   \[
   \frac{3x^3}{x} = 3x^2
   \]
   Multiply \(3x^2\) by \(x + 6\):
   \[
   3x^2(x + 6) = 3x^3 + 18x^2
   \]
   Subtract this from the result of the previous step:
   \[
   (3x^3 + 19x^2 + 4x - 11) - (3x^3 + 18x^2) = x^2 + 4x - 11
   \]

3. **Third Term**: Divide the new first term \(x^2\) by \(x\), giving:
   \[
   \frac{x^2}{x} = x
   \]
   Multiply \(x\) by \(x + 6\):
   \[
   x(x + 6) = x^2 + 6x
   \]
   Subtract this from the result of the previous step:
   \[
   (x^2 + 4x - 11) - (x^2 + 6x) = -2x - 11
   \]

4. **Fourth Term**: Divide the new first term \(-2x\) by \(x\), giving:
   \[
   \frac{-2x}{x} = -2
   \]
   Multiply \(-2\) by \(x + 6\):
   \[
   -2(x + 6) = -2x - 12
   \]
   Subtract this from the result of the previous step:
   \[
   (-2x - 11) - (-2x - 12) = 1
   \]

### Final Answer:
The quotient is:
\[
2x^3 + 3x^2 + x - 2
\]
And the remainder is:
\[
1
\]
So the final answer is:
\[
2x^3 + 3x^2 + x - 2 + \frac{1}{x + 6}
\]

This corresponds to the first option:
\[
\boxed{2x^3 + 3x^2 + x - 2 + \frac{1}{x + 6}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you perform synthetic division for this polynomial?
2. What happens when the remainder is zero in polynomial division?
3. How can you verify the result of a polynomial long division?
4. Can this division problem be solved using factorization instead?
5. What is the degree of the quotient in polynomial division?

### Tip:
Polynomial long division follows the same steps as numerical long division, so keeping track of each step carefully can help avoid mistakes.

Divide (2x^4 + 15x^3 + 19x^2 + 4x - 11) ÷ (x + 6). Write your answer in fraction form.

Learn how to perform polynomial long division for the expression (2x^4 + 15x^3 + 19x^2 + 4x - 11) ÷ (x + 6). This solution provides a step-by-step guide, focusing on the key concepts of polynomial division, algebra, and remainder calculation. Suitable for Grades 10-12 students.

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