Solve the equation -x^2+3x+4<0

1. Rewrite the inequality as x^2 - 3x - 4 > 0.
2. Find the roots of the equation x^2 - 3x - 4 = 0 using the quadratic formula. Here, a = 1, b = -3, and c = -4.

The quadratic formula is:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substitute the values:
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1}

Simplify under the square root:
x = \frac{3 \pm \sqrt{9 + 16}}{2}
x = \frac{3 \pm \sqrt{25}}{2}
x = \frac{3 \pm 5}{2}

This gives the roots:
x = \frac{8}{2} = 4
x = \frac{-2}{2} = -1

3. Use the roots to determine intervals to test:

- Test interval (-\infty, -1).
- Test interval (-1, 4).
- Test interval (4, \infty).

4. Select a test point in each interval and insert it into the inequality:
- In (-\infty, -1), choose x = -2, and substitute into x^2 - 3x - 4:
(-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6 > 0. Satisfies the inequality.

- In (-1, 4), choose x = 0, and substitute into x^2 - 3x - 4:
0^2 - 3(0) - 4 = -4. Does not satisfy the inequality.

- In (4, \infty), choose x = 5, and substitute into x^2 - 3x - 4:
5^2 - 3(5) - 4 = 25 - 15 - 4 = 6 > 0. Satisfies the inequality.

5. The solution is the union of the intervals that satisfy the inequality:
(-\infty, -1) \cup (4, \infty)

Answer: (-\infty, -1) \cup (4, \infty)