**Solving the Equation 2(x^2 + 1) = 5x**

We are given the equation 2(x^2 + 1) = 5x, and we need to find the values of two expressions: (i) x - 1/x and (ii) x^3 - 1/(x^3).

**Step 1: Simplify the Equation**

First, let's simplify the given equation:

2(x^2 + 1) = 5x

Expanding the left-hand side, we get:

2x^2 + 2 = 5x

Subtracting 5x from both sides gives:

2x^2 - 5x + 2 = 0

**Step 2: Factorize the Quadratic Equation**

Now, let's factorize the quadratic equation:

2x^2 - 5x + 2 = (2x - 1)(x - 2) = 0

This gives us two possible values for x:

2x - 1 = 0 => x = 1/2

x - 2 = 0 => x = 2

**Part (i): Find x - 1/x**

Now, let's find the value of x - 1/x for each of the two possible values of x:

**Case 1: x = 1/2**

x - 1/x = (1/2) - 1/(1/2) = (1/2) - 2 = -3/2

**Case 2: x = 2**

x - 1/x = 2 - 1/2 = 3/2

**Part (ii): Find x^3 - 1/(x^3)**

Now, let's find the value of x^3 - 1/(x^3) for each of the two possible values of x:

**Case 1: x = 1/2**

x^3 - 1/(x^3) = (1/2)^3 - 1/(1/2)^3 = (1/8) - 8 = -63/8

**Case 2: x = 2**

x^3 - 1/(x^3) = 2^3 - 1/2^3 = 8 - 1/8 = 63/8

Thus, we have found the values of x - 1/x and x^3 - 1/(x^3) for each of the two possible values of x.