To solve the inequality $x^2 + 3x - 4 > 0$, we can follow these steps:

Step 1: Solve the corresponding quadratic equation

First, solve the equation $x^2 + 3x - 4 = 0$ to find the critical points, which will help us determine the intervals to test for the inequality.

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 3$, and $c = -4$.

Substitute the values into the formula: $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm \sqrt{25}}{2}$ $x = \frac{-3 \pm 5}{2}$ So the solutions are: $x_1 = \frac{-3 + 5}{2} = 1, \quad x_2 = \frac{-3 - 5}{2} = -4$

Step 2: Analyze the sign of the quadratic expression

Now that we know the critical points are $x = -4$ and $x = 1$, we can determine where $x^2 + 3x - 4 > 0$ by testing values in the intervals divided by these critical points: $(-\infty, -4)$, $(-4, 1)$, and $(1, \infty)$.

1. For $x \in (-\infty, -4)$:
   Pick $x = -5$:
   $(-5)^2 + 3(-5) - 4 = 25 - 15 - 4 = 6$ So, the expression is positive.

2. For $x \in (-4, 1)$:
   Pick $x = 0$:
   $0^2 + 3(0) - 4 = -4$ So, the expression is negative.

3. For $x \in (1, \infty)$:
   Pick $x = 2$:
   $2^2 + 3(2) - 4 = 4 + 6 - 4 = 6$ So, the expression is positive.

Step 3: Conclusion

The quadratic expression $x^2 + 3x - 4 > 0$ is positive in the intervals $(-\infty, -4)$ and $(1, \infty)$.

Thus, the solution to the inequality is: $x \in (-\infty, -4) \cup (1, \infty)$

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do we solve inequalities that involve other types of quadratic expressions?
2. What happens when the discriminant of a quadratic equation is zero or negative?
3. How can you graphically represent the solution of a quadratic inequality?
4. What is the significance of critical points in solving inequalities?
5. How do quadratic inequalities differ from linear inequalities?

Tip: Always test values from each interval to determine whether the quadratic expression is positive or negative in that region.