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The inequality provided is:

\[
x^2 + 13x + 42 < 0
\]

To solve this quadratic inequality, follow these steps:

### Step 1: Solve the quadratic equation
We need to find the roots of the equation \(x^2 + 13x + 42 = 0\). This can be done using the quadratic formula:

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

For the quadratic equation \(x^2 + 13x + 42 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 13\)
- \(c = 42\)

Now substitute into the quadratic formula:

\[
x = \frac{-13 \pm \sqrt{13^2 - 4(1)(42)}}{2(1)}
\]

\[
x = \frac{-13 \pm \sqrt{169 - 168}}{2}
\]

\[
x = \frac{-13 \pm \sqrt{1}}{2}
\]

\[
x = \frac{-13 \pm 1}{2}
\]

Thus, the two roots are:

\[
x_1 = \frac{-13 + 1}{2} = \frac{-12}{2} = -6
\]
\[
x_2 = \frac{-13 - 1}{2} = \frac{-14}{2} = -7
\]

### Step 2: Determine the intervals
The quadratic inequality \(x^2 + 13x + 42 < 0\) will be negative between the two roots. Therefore, the intervals of interest are:
- The two roots divide the real line into three intervals: \( (-\infty, -7) \), \( (-7, -6) \), and \( (-6, \infty) \).

### Step 3: Test each interval
We will check the sign of \(x^2 + 13x + 42\) in each interval by choosing a test point:

1. **For \(x \in (-\infty, -7)\), pick \(x = -8\):**
   \[
   (-8)^2 + 13(-8) + 42 = 64 - 104 + 42 = 2 \quad \text{(positive)}
   \]

2. **For \(x \in (-7, -6)\), pick \(x = -6.5\):**
   \[
   (-6.5)^2 + 13(-6.5) + 42 = 42.25 - 84.5 + 42 = -0.25 \quad \text{(negative)}
   \]

3. **For \(x \in (-6, \infty)\), pick \(x = 0\):**
   \[
   0^2 + 13(0) + 42 = 42 \quad \text{(positive)}
   \]

### Step 4: Conclusion
The quadratic expression \(x^2 + 13x + 42\) is negative only in the interval \((-7, -6)\).

Thus, the solution to the inequality is:

\[
x \in (-7, -6)
\]

Let me know if you need any clarifications or further details!

Here are some follow-up questions you could explore:
1. How would the solution change if the inequality were \(\geq 0\)?
2. What is the geometric meaning of the interval \((-7, -6)\) on the graph of the quadratic function?
3. How can you solve quadratic inequalities using the method of completing the square?
4. What happens when the discriminant in the quadratic formula is negative?
5. How do you test intervals for more complex inequalities, such as rational inequalities?

**Tip:** Always check the sign of the quadratic expression in each interval, as this determines where the inequality holds true.

Learn how to solve the quadratic inequality x^2 + 13x + 42 < 0. This solution involves finding the roots using the quadratic formula and testing intervals to determine where the inequality holds. Suitable for Grades 9-10.

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