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To determine when this rational expression is undefined, we need to analyze the denominator of the expression:

\[
\frac{x + 6}{x^2 + x - 12}
\]

A rational expression is undefined when its denominator is equal to zero. So, let's set the denominator equal to zero and solve for \( x \):

\[
x^2 + x - 12 = 0
\]

### Step 1: Factor the Quadratic Expression
We can factor \( x^2 + x - 12 \) as follows:

\[
x^2 + x - 12 = (x + 4)(x - 3)
\]

### Step 2: Set Each Factor to Zero
Now, set each factor equal to zero to find the values of \( x \) that make the expression undefined:

1. \( x + 4 = 0 \Rightarrow x = -4 \)
2. \( x - 3 = 0 \Rightarrow x = 3 \)

### Conclusion
The expression is undefined for \( x = -4 \) and \( x = 3 \). 

Thus, the correct answer is:

**Option A:** The rational expression is undefined for \( x = -4, 3 \).

Would you like further details on any of these steps, or do you have other questions?

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Here are 5 additional questions you may find helpful:

1. How do we handle cases where the denominator is a more complex polynomial?
2. Why is a rational expression undefined when the denominator is zero?
3. How can you check if an expression is defined for all real numbers?
4. What other methods, besides factoring, can we use to solve quadratic equations?
5. How would this change if the numerator was zero instead?

**Tip:** Always remember to check the denominator of a rational expression for values that make it zero to find where the expression is undefined.

Find all numbers for which the rational expression (x + 6) / (x^2 + x - 12) is undefined. If the rational expression is defined for all real numbers, so state.

Discover how to find the values for which the rational expression (x + 6) / (x^2 + x - 12) is undefined. This solution involves factoring quadratic expressions and applying the zero product property, making it ideal for Grades 8-10.

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