https://file.aiquickdraw.com/mathbot/file/1727359736296ap4mv7jj.jpg

To solve the question displayed in the image, we need to divide:

\[
2x^3 + 3x^2 + 5x - 4 \quad \text{by} \quad x^2 + x + 1.
\]

Let's perform the division step by step using polynomial long division:

### Step 1: Divide the first terms
Divide the leading term of the numerator \(2x^3\) by the leading term of the denominator \(x^2\):

\[
\frac{2x^3}{x^2} = 2x.
\]

### Step 2: Multiply and subtract
Multiply \(2x\) by the entire denominator \(x^2 + x + 1\):

\[
2x \cdot (x^2 + x + 1) = 2x^3 + 2x^2 + 2x.
\]

Now, subtract this from the original polynomial:

\[
(2x^3 + 3x^2 + 5x - 4) - (2x^3 + 2x^2 + 2x) = (3x^2 - 2x^2) + (5x - 2x) - 4 = x^2 + 3x - 4.
\]

### Step 3: Divide the new leading term
Now divide the leading term \(x^2\) by \(x^2\):

\[
\frac{x^2}{x^2} = 1.
\]

### Step 4: Multiply and subtract
Multiply \(1\) by the denominator \(x^2 + x + 1\):

\[
1 \cdot (x^2 + x + 1) = x^2 + x + 1.
\]

Now subtract this from the current remainder:

\[
(x^2 + 3x - 4) - (x^2 + x + 1) = (3x - x) - (4 + 1) = 2x - 5.
\]

### Final quotient and remainder
The quotient is \(2x + 1\) with a remainder of \(2x - 5\), which is divided by the original divisor \(x^2 + x + 1\).

So, the result can be written as:

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

### Conclusion
The correct answer is:

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

This matches the third option in the image.

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Do you have any questions or would you like a detailed explanation on any part of this process? 

Here are some related questions to deepen your understanding:

1. How can we verify the result of a polynomial division?
2. What are the steps to perform synthetic division for polynomials?
3. How would the division change if the degrees of the numerator and denominator were different?
4. What is the difference between polynomial long division and synthetic division?
5. How can the remainder theorem be applied to this division?

**Tip:** Always check if you can factor the denominator or simplify before performing polynomial division. This can sometimes make the process easier.

What is the quotient of 2x^3 + 3x^2 + 5x - 4 divided by x^2 + x + 1?

Learn how to perform polynomial long division to find the quotient and remainder of 2x^3 + 3x^2 + 5x - 4 divided by x^2 + x + 1. This detailed solution explains each step in solving this algebra problem, perfect for high school students studying polynomial division.

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