First, we need to define the Muller’s method formula:

x3 = x2 - (2f(x2))/(P ± √(P^2 - 4Qf(x2)))

where P = f(x2) - f(x1)/(x2 - x1) + f(x2) - f(x0)/(x2 - x0)

and Q = (x2 - x1)/(x1 - x0)

Now, let’s plug in the values for our initial guesses:

x0 = 4.5, f(x0) = (4.5)^3 - 13(4.5) - 12 = -38.125

x1 = 5.5, f(x1) = (5.5)^3 - 13(5.5) - 12 = 23.875

x2 = 5, f(x2) = (5)^3 - 13(5) - 12 = -2

P = (-2 - (-38.125))/(5 - 4.5) + (-2 - 23.875)/(5 - 5.5) = -29.25

Q = (5 - 5.5)/(5.5 - 4.5) = -1

Now, we can plug in these values into the Muller’s method formula:

x3 = 5 - (2(-2))/(P ± √(P^2 - 4Qf(5)))

x3 = 5 - (-4)/(P ± √(P^2 + 8))

We need to calculate the two possible values for the denominator:

P + √(P^2 + 8) = -29.25 + √(29.25^2 + 8) = 5.000000000000002

P - √(P^2 + 8) = -29.25 - √(29.25^2 + 8) = -59.00000000000001

We choose the positive value for the denominator, so:

x3 = 5 - (-4)/(5.000000000000002) = 5.8

Now, we need to update our guesses:

x0 = 5.5, x1 = 5, x2 = 5.8

We repeat the process:

x4 = 5.8 - (2f(5.8))/(P ± √(P^2 - 4Qf(5.8)))

x4 = 5.8 - (2(5.8^3 - 13(5.8) - 12))/(P ± √(P^2 - 4Q(-2)))

x4 = 5.8 - (2(107.072))/(P ± √(P^2 + 8))

We calculate the two possible values for the denominator:

P + √(P^2 + 8) = -0.0000000000000004440892098500624

P - √(P^2 + 8) = -118.00000000000001

We choose the positive value for the denominator, so:

x4 = 5.8 - (2(107.072))/(-0.0000000000000004440892098500624) = 4.000000000000001

Now, we need to update our guesses again:

x0 = 5, x1 = 5.8, x2 = 4.000000000000001

We repeat the process:

x5 = 4.000000000000001 - (2f(4.000000000000001))/(P ± √(P^2 - 4Qf(4.000000000000001)))

x5 = 4.000000000000001 - (2(4.000000000000001^3 - 13(4.000000000000001) - 12))/(P ± √(P^2 - 4Q(107.072)))

x5 = 4.000000000000001 - (2(-107.072))/(P ± √(P^2 - 4Q(107.072)))

We calculate the two possible values for the denominator:

P + √(P^2 - 4Q(107.072)) = -0.0000000000000004440892098500624

P - √(P^2 - 4Q(107.072)) = -118.00000000000001

We choose the positive value for the denominator, so:

x5 = 4.000000000000001 - (2(-107.072))/(-0.0000000000000004440892098500624) = -3.000000000000001

We have found a root of the equation F(x) = x^3 - 13x - 12 to be approximately -3.