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The inequality to solve is:

\[
x^2 + 4x - 12 \leq 0
\]

### Steps to Solve:

1. **Find the roots of the quadratic equation:**
   We first solve the related quadratic equation by setting the expression equal to 0:
   
   \[
   x^2 + 4x - 12 = 0
   \]
   
   Using the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   where \(a = 1\), \(b = 4\), and \(c = -12\):

   \[
   x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}
   \]
   \[
   x = \frac{-4 \pm \sqrt{16 + 48}}{2}
   \]
   \[
   x = \frac{-4 \pm \sqrt{64}}{2}
   \]
   \[
   x = \frac{-4 \pm 8}{2}
   \]
   
   So, the two solutions are:
   \[
   x_1 = \frac{-4 + 8}{2} = 2
   \]
   \[
   x_2 = \frac{-4 - 8}{2} = -6
   \]

2. **Determine intervals:**
   The quadratic expression changes sign at \(x = 2\) and \(x = -6\). We now test the sign of the expression \(x^2 + 4x - 12\) in the intervals defined by these roots:
   - \( (-\infty, -6) \)
   - \( (-6, 2) \)
   - \( (2, \infty) \)

3. **Test the intervals:**
   - For \(x = -7\) (in the interval \((- \infty, -6)\)):
     \[
     (-7)^2 + 4(-7) - 12 = 49 - 28 - 12 = 9 > 0
     \]
     So, the expression is positive in this interval.
   
   - For \(x = 0\) (in the interval \((-6, 2)\)):
     \[
     0^2 + 4(0) - 12 = -12 < 0
     \]
     So, the expression is negative in this interval.

   - For \(x = 3\) (in the interval \((2, \infty)\)):
     \[
     3^2 + 4(3) - 12 = 9 + 12 - 12 = 9 > 0
     \]
     So, the expression is positive in this interval.

4. **Conclusion:**
   The quadratic expression is less than or equal to zero in the interval where it is negative or zero. This happens between the roots \(x = -6\) and \(x = 2\).

   Therefore, the solution is:

   \[
   -6 \leq x \leq 2
   \]

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How does the quadratic formula help solve inequalities?
2. What is the graphical interpretation of the quadratic inequality?
3. How do you determine the sign of a quadratic expression between roots?
4. What happens if a quadratic has no real roots?
5. How can this solution be verified by factoring?

**Tip:** Always check the endpoints of an inequality to see if they satisfy the original condition!

Solve the following second-degree inequality: x^2 + 4x - 12 ≤ 0

Learn how to solve the quadratic inequality x^2 + 4x - 12 ≤ 0 with a detailed step-by-step approach. This solution covers algebraic concepts including finding roots using the quadratic formula and analyzing sign intervals. Suitable for students in Grades 9-10.

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