To solve the given polynomial division problem:

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

We'll use polynomial long division.

Steps:

1. Divide the leading term of the numerator by the leading term of the divisor: $\frac{4x^4}{2x^2} = 2x^2$ So, the first term of the quotient is $2x^2$.

2. Multiply $2x^2$ by the divisor $2x^2 - 3$: $(2x^2)(2x^2 - 3) = 4x^4 - 6x^2$

3. Subtract this result from the numerator: $(4x^4 - 4x^2 - x - 3) - (4x^4 - 6x^2) = 2x^2 - x - 3$

4. Divide the new leading term of the remainder by the leading term of the divisor: $\frac{2x^2}{2x^2} = 1$ So, the next term of the quotient is $+1$.

5. Multiply $1$ by $2x^2 - 3$: $(1)(2x^2 - 3) = 2x^2 - 3$

6. Subtract this result from the current remainder: $(2x^2 - x - 3) - (2x^2 - 3) = -x$

7. Since the degree of the remainder ($-x$) is less than the degree of the divisor ($2x^2 - 3$), the division ends here. The remainder is $-x$.

Final Answer:

$\text{Quotient: } 2x^2 + 1 \quad \text{Remainder: } -x$

In terms of the final expression: $\frac{4x^4 - 4x^2 - x - 3}{2x^2 - 3} = 2x^2 + 1 + \frac{-x}{2x^2 - 3}$

If you have more questions or want further clarifications, feel free to ask!

Questions to expand knowledge:

1. What are the differences between polynomial long division and synthetic division?
2. How can the remainder theorem be applied to verify the remainder in this division?
3. What is the role of degrees in determining when the division process ends?
4. Can this process be used with higher-order polynomials? How does it scale?
5. How can you check if your quotient is correct?

Tip: Always align terms by their degrees in long division to avoid errors.