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The problem is asking to find the remainder when the polynomial \(6x^2 + 11x - 3\) is divided by \(x + 2\). We can use **synthetic division** or **polynomial remainder theorem** to solve this.

### Using the **Polynomial Remainder Theorem**:
The remainder when a polynomial \(f(x)\) is divided by \(x - c\) can be found by evaluating \(f(c)\).

Here, the divisor is \(x + 2\), so we need to evaluate the polynomial at \(x = -2\).

1. The polynomial is \(6x^2 + 11x - 3\).
2. Substitute \(x = -2\) into the polynomial:

\[
f(-2) = 6(-2)^2 + 11(-2) - 3
\]
\[
f(-2) = 6(4) + 11(-2) - 3
\]
\[
f(-2) = 24 - 22 - 3
\]
\[
f(-2) = -1
\]

Thus, the remainder is \(-1\).

Would you like further details or have any questions?

### Follow-up Questions:
1. Can you explain how the polynomial remainder theorem works in more depth?
2. How would synthetic division solve the same problem?
3. What is the difference between synthetic division and the polynomial remainder theorem?
4. Could you solve this using long division instead?
5. What would happen if the divisor was a different linear factor, like \(x - 3\)?

### Tip:
Always substitute the opposite sign of the linear divisor when using the polynomial remainder theorem. For \(x + 2\), use \(x = -2\).

What is the remainder when the polynomial 6x^2 + 11x - 3 is divided by x + 2?

Learn how to find the remainder when the polynomial 6x^2 + 11x - 3 is divided by x + 2 using the Polynomial Remainder Theorem. Detailed step-by-step solution suitable for students in Grades 9-12.

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