https://file.aiquickdraw.com/mathbot/file/1726281061043vphfctfm.jpg

How do I express this solution using interval notation?

The inequality in the image is:

\[
x^2 + 5x + 4 > 0
\]

### To solve this inequality:
1. **Factor the quadratic expression** \(x^2 + 5x + 4\):
   \[
   x^2 + 5x + 4 = (x + 4)(x + 1)
   \]
   
2. Now, the inequality becomes:
   \[
   (x + 4)(x + 1) > 0
   \]
   
3. **Find the critical points** by setting the factors equal to 0:
   \[
   x + 4 = 0 \quad \text{or} \quad x + 1 = 0
   \]
   The critical points are:
   \[
   x = -4 \quad \text{and} \quad x = -1
   \]
   
4. **Determine the sign of the expression** on each interval divided by the critical points: \((-∞, -4)\), \((-4, -1)\), and \((-1, ∞)\). To do this, test a point in each interval.

   - For \(x = -5\) in the interval \((-∞, -4)\):
     \[
     (x + 4)(x + 1) = (-5 + 4)(-5 + 1) = (-1)(-4) = 4 > 0
     \]
     So, the expression is positive in \((-∞, -4)\).
   
   - For \(x = -2\) in the interval \((-4, -1)\):
     \[
     (x + 4)(x + 1) = (-2 + 4)(-2 + 1) = (2)(-1) = -2 < 0
     \]
     So, the expression is negative in \((-4, -1)\).
   
   - For \(x = 0\) in the interval \((-1, ∞)\):
     \[
     (x + 4)(x + 1) = (0 + 4)(0 + 1) = 4 > 0
     \]
     So, the expression is positive in \((-1, ∞)\).

5. **Write the solution in interval notation**. Since we want the expression to be positive, the solution is the intervals where the product is positive, excluding the critical points where the product is zero:
   \[
   (-∞, -4) \cup (-1, ∞)
   \]

### Final answer:
The solution in interval notation is:

\[
(-∞, -4) \cup (-1, ∞)
\]

Would you like further details or explanations? Here are some related questions:

1. How do you test intervals for quadratic inequalities?
2. What is the significance of critical points in inequalities?
3. Can you solve a similar inequality but with \(< 0\) instead?
4. How can we represent the solution graphically?
5. How would the solution change if it were a non-strict inequality (\(\geq 0\))?

**Tip:** Always remember that the critical points are excluded for strict inequalities (like \(>\) or \(<\)).

Learn how to solve the quadratic inequality x^2 + 5x + 4 > 0 by factoring the quadratic expression and finding the solution using interval notation. Suitable for students in Grades 9-10.

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