https://file.aiquickdraw.com/mathbot/file/1729587830591xkbch1e6.jpg

The expression in the image is:

\[
\frac{2x^3 + 5x^2 + x - 1}{x + 2}
\]

This can be simplified using polynomial division or synthetic division. Let's use synthetic division, which is faster for dividing by linear factors like \(x + 2\).

### Synthetic Division Steps:

1. **Set up the synthetic division** by using the root of \(x + 2\), which is \(x = -2\).
2. The coefficients of the polynomial \(2x^3 + 5x^2 + x - 1\) are:
   \[
   [2, 5, 1, -1]
   \]

3. Now, perform the synthetic division:

- Bring down the leading coefficient \(2\).
- Multiply \(2\) by \(-2\) to get \(-4\), then add to \(5\): \(5 + (-4) = 1\).
- Multiply \(1\) by \(-2\) to get \(-2\), then add to \(1\): \(1 + (-2) = -1\).
- Multiply \(-1\) by \(-2\) to get \(2\), then add to \(-1\): \(-1 + 2 = 1\).

So, the quotient is:
\[
2x^2 + x - 1
\]
and the remainder is \(1\).

### Final Result:

The division yields:
\[
\frac{2x^3 + 5x^2 + x - 1}{x + 2} = 2x^2 + x - 1 + \frac{1}{x + 2}
\]

This is the simplified form of the given expression.

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Would you like a deeper explanation or more details on any part of this solution?

### 5 Related Questions:
1. How do you perform polynomial long division?
2. What is synthetic division and when is it most useful?
3. Can all polynomials be divided using synthetic division?
4. How do you interpret the remainder in polynomial division?
5. What is the difference between polynomial division and factorization?

### Tip:
Synthetic division is a faster method for dividing polynomials when the divisor is of the form \(x - a\). Always ensure the polynomial is arranged in descending powers before applying this method.

Learn how to simplify the polynomial expression (2x^3 + 5x^2 + x - 1) ÷ (x + 2) using synthetic division. This problem covers key algebraic concepts such as polynomial division and the use of synthetic division for linear factors. Suitable for students in Grades 10-12.

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