## Answer :

### Step-by-Step Analysis:

1.

**Form of the Function**:

The function is given as [tex]\( f(x) = (x+4)(x-6) \)[/tex].

2.

**Expansion of the Function**:

Expanding [tex]\( f(x) \)[/tex]:

[tex]\[ f(x) = (x+4)(x-6) = x^2 - 2x - 24 \][/tex]

3.

**Vertex of the Function**:

The vertex form of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] gives us the x-coordinate of the vertex as [tex]\( x = -\frac{b}{2a} \)[/tex].

Given [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -24 \)[/tex]:

[tex]\[ x_{\text{vertex}} = -\frac{-2}{2 \cdot 1} = 1 \][/tex]

To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] back into the function:

[tex]\[ y_{\text{vertex}} = (1^2) - 2(1) - 24 = 1 - 2 - 24 = -25 \][/tex]

Therefore, the vertex of the function is at [tex]\( (1, -25) \)[/tex].

4.

**Intervals Where the Function is Increasing/Decreasing**:

Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards. This means:

- The function is

**decreasing**to the left of the vertex (i.e., for [tex]\( x < 1 \)[/tex])

- The function is

**increasing**to the right of the vertex (i.e., for [tex]\( x > 1 \)[/tex])

5.

**Intervals Where the Function Is Positive/Negative**:

Finding the roots of the quadratic equation [tex]\( x^2 - 2x - 24 = 0 \)[/tex]:

[tex]\[ (x+4)(x-6) = 0 \][/tex]

Thus, the solutions are [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].

- The function is

**positive**on the intervals [tex]\( (-\infty, -4) \)[/tex] and [tex]\( (6, \infty) \)[/tex]

- The function is

**negative**on the interval [tex]\( (-4, 6) \)[/tex]

### Conclusion:

Given the above analysis, the correct statements are:

1.

**The vertex of the function is at [tex]\( (1, -25) \)[/tex]**. This is a true statement, as determined from the vertex calculation.

2.

**The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex]**. This is true because the function is below the x-axis (negative) between its roots [tex]\( x = -4 \)[/tex] and [tex]\( x = 6 \)[/tex].

The other statements do not correctly describe the properties of the function.

### Answer:

- The vertex of the function is at [tex]\( (1, -25) \)[/tex].

- The graph is negative on the entire interval [tex]\( -4 < x < 6 \)[/tex].