-x²-4x+5>0

To solve the inequality -x² - 4x + 5 > 0, we need to find the values of x that satisfy the inequality. This is a quadratic inequality, and we can solve it by finding the roots of the corresponding quadratic equation -x² - 4x + 5 = 0.

First, let's find the roots of the quadratic equation. We can do this by using the quadratic formula: x = (-b ± √(b² - 4ac))/(2a), where the quadratic equation is in the form ax² + bx + c = 0.

In this case, the quadratic equation is -x² - 4x + 5 = 0, so a = -1, b = -4, and c = 5.

The roots are given by: x = (-(-4) ± √((-4)² - 4(-1)(5)))/(2(-1)) x = (4 ± √(16 + 20))/(-2) x = (4 ± √36)/(-2) x = (4 ± 6)/(-2)

So, the roots are x = 5 and x = -1.

Now, let's analyze the inequality -x² - 4x + 5 > 0 in relation to these roots.

1. Choose a test point from each of the three intervals created by the roots (-∞, -1), (-1, 5), and (5, +∞). For example, you can choose -2, 0, and 6.

2. Substitute these test points into the inequality -x² - 4x + 5 > 0 to determine the sign of the expression in each interval:

For x = -2: -(-2)² - 4(-2) + 5 = -4 + 8 + 5 = 9, which is positive. For x = 0: -(0)² - 4(0) + 5 = 5, which is positive. For x = 6: -6² - 4(6) + 5 = -36 - 24 + 5 = -55, which is negative.

Therefore, the inequality -x² - 4x + 5 > 0 is satisfied when x is in the interval (-1, 5).

In interval notation, the solution to the inequality is (-1, 5).