The given expression is:

$\frac{x^3 - 2x^2 - 4}{x - 2}$

To evaluate this, we need to perform polynomial division (synthetic division or long division). Let's divide $x^3 - 2x^2 - 4$ by $x - 2$.

Step 1: Perform synthetic division

The division of the polynomial can be done using synthetic division, but let's first try long division:

1. Divide the leading term of $x^3$ by the leading term of $x$, which gives $x^2$.
2. Multiply $x^2$ by $x - 2$, which gives $x^3 - 2x^2$.
3. Subtract $x^3 - 2x^2$ from $x^3 - 2x^2 - 4$, resulting in $0 - 4 = -4$.

At this point, the remainder is $-4$, and the quotient is $x^2$. Therefore:

$\frac{x^3 - 2x^2 - 4}{x - 2} = x^2 - \frac{4}{x - 2}$

This matches option d in the question.

Answer:

The correct option is d.

Would you like a more detailed explanation or have any further questions?

Here are some related questions:

1. What is synthetic division and how does it work?
2. Can we solve similar expressions with different divisors?
3. What happens if the remainder is not zero in polynomial division?
4. How do we factor polynomials to simplify expressions?
5. What are some common errors when performing polynomial division?

Tip: Synthetic division is usually quicker than long division, especially for linear divisors like $x - 2$.