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Let's solve each of the given rational expressions step by step.

### 1. Expression 1:

\[
\frac{x^2 + 8x + 15}{x + 5}
\]

**Factor the numerator**:
- \(x^2 + 8x + 15 = (x + 5)(x + 3)\)

Thus, the expression simplifies to:

\[
\frac{(x + 5)(x + 3)}{x + 5}
\]

Cancel \(x + 5\) from the numerator and denominator (assuming \(x \neq -5\)):

\[
x + 3
\]

**Simplified Result**: \(x + 3\)

---

### 2. Expression 2:

\[
\frac{2x^3 + 7x^2 + 10x + 8}{x + 2}
\]

Here, perform synthetic division since the degree of the numerator is greater than the denominator. I'll use synthetic division to divide \(2x^3 + 7x^2 + 10x + 8\) by \(x + 2\):

\[
\text{Quotient: } 2x^2 + 3x + 4 \quad \text{and remainder: } 0
\]

Thus, the expression simplifies to:

\[
2x^2 + 3x + 4
\]

**Simplified Result**: \(2x^2 + 3x + 4\)

---

### 3. Expression 3:

\[
\frac{x^2 - 3x + 6}{x + 2}
\]

Attempt synthetic division for this expression:

\[
\text{Quotient: } x - 5 \quad \text{and remainder: } -4
\]

Thus, the expression can be written as:

\[
x - 5 + \frac{-4}{x + 2}
\]

**Simplified Result**: \(x - 5 - \frac{4}{x + 2}\)

---

### 4. Expression 4:

\[
\frac{x^5 + x^2 + x^4 + x^3 + x + 1}{x - 1}
\]

We perform synthetic division for this expression as well:

\[
\text{Quotient: } x^4 + 2x^3 + 3x^2 + 4x + 5 \quad \text{and remainder: } 6
\]

Thus, the expression becomes:

\[
x^4 + 2x^3 + 3x^2 + 4x + 5 + \frac{6}{x - 1}
\]

**Simplified Result**: \(x^4 + 2x^3 + 3x^2 + 4x + 5 + \frac{6}{x - 1}\)

---

### 5. Expression 5:

\[
\frac{x^8 + x^7 + x^4 + x^5 + x^6 + x^3 + x^2 + x}{x + 1}
\]

After arranging and combining like terms, the expression becomes:

\[
\frac{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x}{x + 1}
\]

Performing synthetic division:

\[
\text{Quotient: } x^7 \quad \text{and remainder: } 0
\]

Thus, the expression simplifies to:

\[
x^7
\]

**Simplified Result**: \(x^7\)

---

Let me know if you would like more details or have any questions! Here are five follow-up questions you could explore:

1. What is synthetic division, and how does it simplify polynomial division?
2. Can all rational expressions be simplified using factoring?
3. How would the simplifications change if different values were plugged into the variable \(x\)?
4. What would happen if the denominator had higher powers or more complex expressions?
5. Could these techniques be applied to expressions with complex numbers?

**Tip**: When simplifying rational expressions, always check if the denominator equals zero for specific values of \(x\) to avoid undefined results.

Simplify each of the following rational expressions: 
1. (x^2 + 8x + 15) / (x + 5) 
2. (2x^3 + 7x^2 + 10x + 8) / (x + 2) 
3. (x^2 - 3x + 6) / (x + 2) 
4. (x^5 + x^2 + x^4 + x^3 + x + 1) / (x - 1) 
5. (x^8 + x^7 + x^4 + x^5 + x^6 + x^3 + x^2 + x) / (x + 1)

Simplify complex rational expressions, including factorization and polynomial division. Learn how to solve rational expressions like (x^2 + 8x + 15) / (x + 5) using algebraic techniques, synthetic division, and the Factor Theorem. Suitable for high school students in Grades 10-12.

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