-x²-4x+5>0

To solve the inequality \( -x^2 - 4x + 5 > 0 \), we can follow these steps:

1. First, let's rewrite the inequality in the form of a quadratic equation set equal to zero: \( -x^2 - 4x + 5 = 0 \).
2. Next, let's find the roots of the quadratic equation by solving the equation \( -x^2 - 4x + 5 = 0 \).
3. Finally, we will determine the intervals where the inequality is satisfied based on the solutions of the quadratic equation.

Solving the quadratic equation \( -x^2 - 4x + 5 = 0 \):

The quadratic equation can be factored as follows: \[ -x^2 - 4x + 5 = -(x^2 + 4x - 5) = - (x - 1)(x - 5) = 0 \]

The roots of the equation are x = 1 and x = 5.

Now, our inequality can be rewritten as \( -x^2 - 4x + 5 > 0 \) can be analyzed in relation to the roots:

1. \( x < 1 \): Substitute x = 0, \( -0^2 - 4*0 + 5 = 5 > 0 \)
2. \( 1 < x < 5 \): Substitute x = 3, \( -3^2 - 4*3 + 5 = -9 - 12 + 5 = -16 < 0 \)
3. \( x > 5 \): Substitute x = 6, \( -6^2 - 4*6 + 5 = -36 - 24 + 5 = -55 < 0 \)

Therefore, the solution to the inequality \( -x^2 - 4x + 5 > 0 \) is \( x < 1 \) or \( x > 5 \).