The expression you've provided is:

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

To simplify this expression, we can try to factor the numerator if possible. However, let's check if the numerator can be factored in a way that includes \(x + 3\) as one of the factors.

### Step 1: Factoring the Numerator
We need to determine if the quadratic \(x^2 + 5x + 5\) can be factored. Unfortunately, it does not factor nicely over the integers, so we'll proceed with polynomial division.

### Step 2: Polynomial Division
We'll perform polynomial division of \(x^2 + 5x + 5\) by \(x + 3\):

1. **Divide \(x^2\) by \(x\)** to get \(x\).
2. **Multiply \(x\) by \(x + 3\)** to get \(x^2 + 3x\).
3. **Subtract** \(x^2 + 3x\) from \(x^2 + 5x + 5\) to get \(2x + 5\).
4. **Divide \(2x\) by \(x\)** to get \(2\).
5. **Multiply \(2\) by \(x + 3\)** to get \(2x + 6\).
6. **Subtract** \(2x + 6\) from \(2x + 5\) to get \(-1\).

The result of the division is:

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

So, the expression simplifies to:

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

### Final Answer

The simplified expression is:

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

Would you like more details on any of these steps or have any questions?

Here are some related questions:
1. How do you perform polynomial division with different degrees?
2. Can the quadratic formula be applied directly to factor quadratic expressions?
3. What are common methods to simplify rational expressions?
4. How does polynomial division relate to synthetic division?
5. What is the significance of remainders in polynomial division?

**Tip:** When simplifying rational expressions, always check if the numerator can be factored to cancel out the denominator, making the expression simpler.

Learn how to simplify the rational expression x^2 + 5x + 5 / (x + 3) using polynomial division. This problem involves concepts of polynomial division and rational expressions, suitable for high school students.

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