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To determine the domain restrictions for the rational expression 

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

we need to identify values of \( x \) that would make the denominator zero, as division by zero is undefined.

1. Start with the denominator:
   \[
   x^2 + 2x - 3 = 0
   \]

2. Factor the quadratic expression:
   \[
   x^2 + 2x - 3 = (x + 3)(x - 1) = 0
   \]

3. Set each factor equal to zero to solve for \( x \):
   \[
   x + 3 = 0 \Rightarrow x = -3
   \]
   \[
   x - 1 = 0 \Rightarrow x = 1
   \]

Therefore, the values that make the denominator zero are \( x = -3 \) and \( x = 1 \). These are the values that must be excluded from the domain.

### Answer
The correct domain restrictions for this rational expression are:
- \( x \neq -3 \)
- \( x \neq 1 \)

### Correct Options
You should select:
- \( x \neq -3 \)
- \( x \neq 1 \)

Would you like a further breakdown, or do you have any questions?

### Related Questions
1. What are domain restrictions in rational expressions, and why are they important?
2. How do you factor quadratic expressions like \( x^2 + 2x - 3 \)?
3. Can the numerator affect the domain restrictions of a rational expression?
4. What if there were additional terms in the denominator? How would that affect the solution?
5. How can you verify the factors of a quadratic polynomial?

### Tip
When factoring, check each factor by expanding it back to ensure accuracy in simplifying expressions.

Which domain restrictions apply to the rational expression (x^2 + x + 6) / (x^2 + 2x - 3)?

Learn how to determine the domain restrictions for the rational expression (x^2 + x + 6) / (x^2 + 2x - 3) by factoring the denominator and applying the zero-product property. This problem covers key algebra concepts and is suitable for students in Grades 9-10.

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