You want to divide the polynomial \( \frac{x^3 + 2x^2 - 5x + 4}{x - 3} \). This can be done using polynomial long division. Let's go through the steps:

### Step 1: Set up the long division
We'll divide \( x^3 + 2x^2 - 5x + 4 \) by \( x - 3 \).

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

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

### Step 3: Multiply and subtract
Now, multiply \( x^2 \) by \( x - 3 \):

\[
x^2(x - 3) = x^3 - 3x^2
\]

Subtract this from the original polynomial:

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

### Step 4: Repeat the process
Now divide \( 5x^2 \) by \( x \):

\[
\frac{5x^2}{x} = 5x
\]

Multiply \( 5x \) by \( x - 3 \):

\[
5x(x - 3) = 5x^2 - 15x
\]

Subtract this from the new polynomial:

\[
(5x^2 - 5x + 4) - (5x^2 - 15x) = 10x + 4
\]

### Step 5: Continue the process
Now divide \( 10x \) by \( x \):

\[
\frac{10x}{x} = 10
\]

Multiply \( 10 \) by \( x - 3 \):

\[
10(x - 3) = 10x - 30
\]

Subtract this from the current polynomial:

\[
(10x + 4) - (10x - 30) = 34
\]

### Step 6: Write the result
The quotient is:

\[
x^2 + 5x + 10
\]

And the remainder is:

\[
34
\]

Thus, the result of the division is:

\[
\frac{x^3 + 2x^2 - 5x + 4}{x - 3} = x^2 + 5x + 10 + \frac{34}{x - 3}
\]

Would you like more details on any part of this process?

### Related questions:
1. How do you perform synthetic division?
2. Can you use polynomial long division for higher-degree polynomials?
3. What is the difference between polynomial long division and synthetic division?
4. How do you check if a polynomial division is correct?
5. How do you factor higher-degree polynomials?

### Tip:
Always check if you can factor the polynomial before attempting long division—it may simplify the process!

Learn how to perform polynomial long division for the expression (x^3 + 2x^2 - 5x + 4) divided by (x - 3). This detailed solution breaks down each step, covering key algebraic concepts like polynomial division and the division algorithm for polynomials. Suitable for high school students in Grades 9-12.

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