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The inequality provided is:

\[
x^2 < 3x + 18
\]

### Steps to solve:

1. **Rewrite the inequality**: 
   Move all terms to one side of the inequality:
   \[
   x^2 - 3x - 18 < 0
   \]

2. **Factor the quadratic expression**:
   We need to factor the quadratic \( x^2 - 3x - 18 \). First, find two numbers that multiply to \(-18\) and add up to \(-3\). These numbers are 3 and \(-6\):
   \[
   (x - 6)(x + 3) < 0
   \]

3. **Find critical points**:
   The critical points occur when the factors are equal to zero:
   \[
   x - 6 = 0 \quad \text{or} \quad x + 3 = 0
   \]
   Solving these gives the critical points \( x = 6 \) and \( x = -3 \).

4. **Test intervals**:
   The critical points divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 6) \), and \( (6, \infty) \). Test a point in each interval to determine where the inequality holds true.

   - For \( x = -4 \) (in \( (-\infty, -3) \)): 
     \[
     (-4 - 6)(-4 + 3) = (-10)(-1) = 10 > 0 \quad (\text{False})
     \]
   - For \( x = 0 \) (in \( (-3, 6) \)):
     \[
     (0 - 6)(0 + 3) = (-6)(3) = -18 < 0 \quad (\text{True})
     \]
   - For \( x = 7 \) (in \( (6, \infty) \)):
     \[
     (7 - 6)(7 + 3) = (1)(10) = 10 > 0 \quad (\text{False})
     \]

5. **Solution in interval notation**:
   The inequality is satisfied on the interval \( (-3, 6) \).

### Final Answer:
The solution set is \( (-3, 6) \).

Would you like more details on any step? Here are 5 related questions to deepen your understanding:

1. How do you graph the solution to a quadratic inequality on a number line?
2. What other methods can be used to solve quadratic inequalities besides factoring?
3. How do the critical points relate to the solution of polynomial inequalities?
4. How would the solution change if the inequality were \( x^2 \leq 3x + 18 \)?
5. What happens if we can't factor the quadratic equation easily?

**Tip**: Always check the intervals by testing values to ensure the inequality holds true within those ranges.

Solve the following polynomial inequality: x^2 < 3x + 18

Learn how to solve the quadratic inequality x^2 < 3x + 18. This problem involves key algebraic concepts such as factoring and quadratic inequalities, suitable for students in Grades 9-10.

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