If (x + 4) is a factor of x^3 - 14x - 24, we can use the factor theorem to find the other factors.

Factor Theorem: If (x - a) is a factor of a polynomial f(x), then f(a) = 0.

In this case, (x + 4) is a factor, so let's substitute -4 for x in the polynomial:

f(-4) = (-4)^3 - 14(-4) - 24
= -64 + 56 - 24
= -32

Since f(-4) is not equal to 0, (x + 4) is not a factor of the polynomial. Therefore, we need to find the other factors.

Synthetic Division: We can use synthetic division to divide the polynomial by (x + 4) and find the remaining factors.

Let's perform synthetic division:

-4 | 1 0 -14 -24
| -4 16 -8
|-------------------
1 -4 2 -32

The result of the synthetic division is 1x^2 - 4x + 2 with a remainder of -32.

Quadratic Equation: Now, we have a quadratic equation 1x^2 - 4x + 2 - 32 = 0. To find the other factors, we can solve this quadratic equation using the quadratic formula.

The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 1, b = -4, and c = -30. Substituting these values into the quadratic formula:

x = (4 ± √((-4)^2 - 4(1)(-30))) / 2(1)
= (4 ± √(16 + 120)) / 2
= (4 ± √136) / 2
= (4 ± 2√34) / 2
= 2 ± √34

Therefore, the other factors of the polynomial x^3 - 14x - 24 are (x - 2 + √34) and (x - 2 - √34).

Summary:
- The factor (x + 4) is not a factor of the polynomial x^3 - 14x - 24.
- Using synthetic division, we found the remaining quadratic factor as 1x^2 - 4x + 2 with a remainder of -32.
- By solving the quadratic equation, we obtained the other factors as (x - 2 + √34) and (x - 2 - √34).

Use factor theorem