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Let's break down the given problem step-by-step.

We are tasked with dividing the expression:

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

### Step 1: Rearrange the terms in descending order of powers of \(x\)

The given polynomial is written as:

\[
5x^3 + x^2 - 1
\]

Now, we will perform polynomial long division, dividing this expression by \(x + 3\).

### Step 2: Polynomial Long Division

1. Divide the leading term of the dividend (the highest degree term, \(5x^3\)) by the leading term of the divisor (\(x\)):
   \[
   \frac{5x^3}{x} = 5x^2
   \]
   Multiply \(5x^2\) by \(x + 3\):
   \[
   5x^2(x + 3) = 5x^3 + 15x^2
   \]
   Subtract this from the original polynomial:
   \[
   (5x^3 + x^2 - 1) - (5x^3 + 15x^2) = -14x^2 - 1
   \]

2. Divide the leading term of the new polynomial \(-14x^2\) by the leading term of the divisor \(x\):
   \[
   \frac{-14x^2}{x} = -14x
   \]
   Multiply \(-14x\) by \(x + 3\):
   \[
   -14x(x + 3) = -14x^2 - 42x
   \]
   Subtract this from the current remainder:
   \[
   (-14x^2 - 1) - (-14x^2 - 42x) = 42x - 1
   \]

3. Divide the leading term of \(42x\) by the leading term \(x\):
   \[
   \frac{42x}{x} = 42
   \]
   Multiply \(42\) by \(x + 3\):
   \[
   42(x + 3) = 42x + 126
   \]
   Subtract this from the current remainder:
   \[
   (42x - 1) - (42x + 126) = -127
   \]

The quotient so far is \(5x^2 - 14x + 42\) with a remainder of \(-127\). Therefore, the division gives:

\[
5x^2 - 14x + 42 + \frac{-127}{x + 3}
\]

### Step 3: Compare with the provided answer

The provided solution is \(x^2 + 2x - 6 + \frac{17}{x + 3}\), but our result is quite different. This suggests that either the problem was misunderstood or there's an error in the division process.

Would you like a more detailed breakdown of the steps, or should I double-check specific parts of the division?

### Related Questions:
1. What is the purpose of polynomial long division in algebra?
2. How do you handle remainders in polynomial division?
3. What other methods exist to divide polynomials apart from long division?
4. How can synthetic division simplify the process when dividing by a linear binomial?
5. Can polynomial division help in factoring polynomials?

#### Tip:
When performing polynomial division, keep a close eye on the signs to avoid common errors in subtraction!

Divide the expression: \( \frac{x^2 + 5x^3 - 1}{x + 3} \). Compare with the result: \( x^2 + 2x - 6 + \frac{17}{x + 3} \).

Learn how to divide the polynomial expression \(x^2 + 5x^3 - 1\) by \(x + 3\) using polynomial long division. Understand the step-by-step process and compare the result with the proposed solution: \(x^2 + 2x - 6 + \frac{17}{x + 3}\). Suitable for high school students in Grades 9-12.

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