Use synthetic division to find the result when 2, x, cubed, plus, 3, x, squared, minus, 11, x, minus, 10x 
3
 +3x 
2
 −11x−10 is divided by x, plus, 3x+3. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​
 .

The question appears to contain some typographical issues, but I understand you're asking for synthetic division to divide a cubic polynomial by another binomial of the form \(x + 3\). Here's how we'll approach it:

**Given polynomial**:
\[ 2x^3 + 3x^2 - 11x - 10 \]

**Divisor**:
\[ x + 3 \]

We can proceed by using **synthetic division**. Synthetic division works only when dividing by binomials of the form \(x - c\), so we will rewrite \(x + 3\) as \(x - (-3)\), where \(c = -3\).

### Steps for Synthetic Division:

1. **Set up the synthetic division table**:  
   - Coefficients of the dividend: \(2\), \(3\), \(-11\), and \(-10\).
   - The divisor is \(x + 3\), so we use \(c = -3\).

2. **Perform the synthetic division**:
   
\[
\begin{array}{r|rrrr}
-3 & 2 & 3 & -11 & -10 \\
   &   & -6 & 9  & 6  \\
\hline
   & 2 & -3 & -2  & -4 \\
\end{array}
\]

### Explanation:

- Bring down the first coefficient \(2\).
- Multiply \(2\) by \(-3\) to get \(-6\), and add it to the second coefficient \(3\), yielding \(-3\).
- Multiply \(-3\) by \(-3\) to get \(9\), and add it to the third coefficient \(-11\), yielding \(-2\).
- Multiply \(-2\) by \(-3\) to get \(6\), and add it to the last coefficient \(-10\), yielding \(-4\).

### Result:
The quotient is \(2x^2 - 3x - 2\), and the remainder is \(-4\).

### Final Answer:
The result of the division is:
\[
q(x) = 2x^2 - 3x - 2
\]
\[
r(x) = -4
\]
So, the division result is:
\[
2x^2 - 3x - 2 + \frac{-4}{x + 3}
\]

Would you like further details on synthetic division or any other step?

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Here are five related questions:
1. How do you handle synthetic division for polynomials of higher degrees?
2. What is the difference between synthetic and long division in polynomials?
3. How do you factor polynomials after performing synthetic division?
4. How do you find roots using synthetic division?
5. What happens if the divisor is not a linear binomial like \(x + 3\)?

**Tip:** Synthetic division is faster than long division for binomials but works only for linear divisors (degree 1).

Use synthetic division to find the result when 2x^3 + 3x^2 - 11x - 10 is divided by x + 3. If there is a remainder, express the result in the form q(x) + r(x) / b(x).

Learn how to use synthetic division to divide the polynomial 2x^3 + 3x^2 - 11x - 10 by x + 3, including a step-by-step solution with detailed explanations of the remainder theorem. This problem is ideal for students in Grades 9-12 studying algebra and polynomials.

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