Given information:
- The polynomial f(x) is a polynomial of degree four.
- The coefficient of x^4 is 1.
- The values of f(-2), f(2), f(3), and f(4) are known.

Approach:
To find the value of f(5), we need to determine the equation of the polynomial f(x) based on the given information. Then we can substitute x = 5 into the equation to find the value of f(5).

Step 1: Finding the polynomial equation:
Let's assume the polynomial equation to be f(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are constants.

Since the coefficient of x^4 is 1, we can rewrite the equation as f(x) = x^4 + bx^3 + cx^2 + dx + e.

Step 2: Substituting the given values:
Using the given information, we can substitute the x-values and the corresponding f(x)-values into the equation to form a system of equations.

1) f(-2) = -6:
(-2)^4 + b(-2)^3 + c(-2)^2 + d(-2) + e = -6
16 - 8b + 4c - 2d + e = -6

2) f(2) = 6:
(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 6
16 + 8b + 4c + 2d + e = 6

3) f(3) = 9:
(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 9
81 + 27b + 9c + 3d + e = 9

4) f(4) = 12:
(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 12
256 + 64b + 16c + 4d + e = 12

Step 3: Solving the system of equations:
Solving the system of equations formed in step 2 will give us the values of the constants a, b, c, d, and e.

Adding equations 1 and 2:
32 + 16c + 4e = 0 (Equation 5)

Subtracting equation 1 from equation 2:
2b + 2d = 12 (Equation 6)

Subtracting equation 3 from equation 4:
37 + 37b + 7c + d = 3 (Equation 7)

Subtracting equation 3 from equation 5:
51 + 23c + 7e = -9 (Equation 8)

Using equations 6 and 7, we can solve for b and d:
2b = 12 - 2d
b = 6 - d

Substituting the value of b in equation 7:
37 + 37(6 - d) + 7c + d = 3
222 - 37d +