Solve the Quadratic Inequality: When is -x² + 4x - 5 < 0?

Let's analyze the function $y = -x^2 + 4x - 5$ and determine the interval where $y$ is negative.

1. **Find the roots using the quadratic formula**:
The function is given by $y = -x^2 + 4x - 5$. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = -1$, $b = 4$, and $c = -5$. First, we calculate the discriminant:

$b^2 - 4ac = 4^2 - 4(-1)(-5) = 16 - 20 = -4$

Since the discriminant is negative, the quadratic equation has no real roots, implying that the parabola does not intersect the x-axis. The quadratic formula confirms there are no real solutions, confirming the function does not touch or cross the x-axis.

2. **Analyze the parabola's direction**:
Since $a = -1$, the parabola opens downwards. A downward-opening parabola with no real roots means it lies entirely below the x-axis. Hence, the function $y = -x^2 + 4x - 5$ is negative for all values of $x$.

Therefore, the function is negative for all $x$.