We are asked to solve the inequality:

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

### Step 1: Solve the related equation \(x^2 + 5x + 4 = 0\)

To solve this, we factor the quadratic expression:

\[
x^2 + 5x + 4 = (x + 1)(x + 4) = 0
\]

Thus, the roots are:

\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

### Step 2: Analyze the sign of the inequality

We now consider the intervals determined by the roots \((- \infty, -4)\), \((-4, -1)\), and \((-1, \infty)\).

- For \(x < -4\) (choose \(x = -5\)):
  \[
  (-5 + 1)(-5 + 4) = (-4)(-1) = 4 > 0
  \]
  The expression is positive in this interval.

- For \(-4 < x < -1\) (choose \(x = -2\)):
  \[
  (-2 + 1)(-2 + 4) = (-1)(2) = -2 < 0
  \]
  The expression is negative in this interval.

- For \(x > -1\) (choose \(x = 0\)):
  \[
  (0 + 1)(0 + 4) = (1)(4) = 4 > 0
  \]
  The expression is positive in this interval.

### Step 3: Conclusion

The inequality \(x^2 + 5x + 4 > 0\) is satisfied when the expression is positive, which occurs in the intervals \((- \infty, -4)\) and \((-1, \infty)\). Therefore, the solution to the inequality is:

\[
x \in (-\infty, -4) \cup (-1, \infty)
\]

Let me know if you'd like further details or clarification!

Here are 5 related questions:
1. How do you solve quadratic inequalities with different coefficients?
2. Can the quadratic formula be used to solve inequalities directly?
3. What changes when the inequality is \(x^2 + 5x + 4 \geq 0\)?
4. How do you test intervals for the sign of a quadratic function?
5. What happens if the quadratic equation has complex roots?

**Tip:** Always factor the quadratic expression first, if possible, as it simplifies solving inequalities significantly.

Learn how to solve the quadratic inequality x^2 + 5x + 4 > 0 using factoring and sign analysis. This detailed solution explores intervals, roots, and positive/negative regions of the function. Perfect for students in Grades 8-10 studying algebra and inequalities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation