Let's analyze the image and the problem.

The quadratic function given is:

$f(x) = x^2 + 4x - 5$

We need to answer a few parts based on this function:

1. Determine where the function is increasing and decreasing.

   To find where the function is increasing or decreasing, we start by finding the vertex of the quadratic function.

   For a quadratic function $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by: $x = \frac{-b}{2a}$

   Here, $a = 1$ and $b = 4$, so: $x = \frac{-4}{2(1)} = -2$

   Therefore, the vertex occurs at $x = -2$. A quadratic function is:

   • Decreasing on the left of the vertex ($x < -2$)
   • Increasing on the right of the vertex ($x > -2$)

   Hence, the function is:

   • Decreasing on the interval $(-\infty, -2]$
   • Increasing on the interval $[-2, \infty)$

2. Determine where $f(x) > 0$ and where $f(x) < 0$.

   To find this, we first need to find the roots of the quadratic function by solving $f(x) = 0$: $x^2 + 4x - 5 = 0$ Factorizing: $(x + 5)(x - 1) = 0$ So, the roots are $x = -5$ and $x = 1$.

   Now, analyze the sign of the function:

   • $f(x) > 0$ when $x < -5$ or $x > 1$
   • $f(x) < 0$ when $-5 < x < 1$

3. Based on the intervals, the answer to part (f) is:

   • Option B: $f(x) > 0$ on $(-\infty, -5) \cup (1, \infty)$ and $f(x) < 0$ on $(-5, 1)$.

Let me know if you'd like further details or clarification!

Here are some related questions:

1. How do you find the axis of symmetry for a quadratic function?
2. What is the significance of the vertex in a quadratic function?
3. How can you solve quadratic equations by completing the square?
4. What is the discriminant of a quadratic equation, and what does it tell us?
5. How do you graph a quadratic function step by step?

Tip: When analyzing quadratic functions, always start by finding the vertex to determine where the function increases or decreases!