To divide \(x^5 - 4x^2 - 5\) by \(x^2 - 5\), we can use polynomial long division.

### Step-by-Step Solution:

1. **Setup the Division:**

   \[
   \text{Divide } x^5 - 4x^2 - 5 \text{ by } x^2 - 5.
   \]

2. **Divide the Leading Terms:**

   The leading term of the dividend \(x^5\) divided by the leading term of the divisor \(x^2\) gives:
   \[
   x^5 \div x^2 = x^3.
   \]

   So, the first term in the quotient is \(x^3\).

3. **Multiply and Subtract:**

   Multiply the entire divisor \(x^2 - 5\) by \(x^3\) and subtract it from the original polynomial:
   
   \[
   (x^5 - 4x^2 - 5) - (x^3(x^2 - 5)) = (x^5 - 4x^2 - 5) - (x^5 - 5x^3).
   \]

   Simplifying, we get:
   
   \[
   x^5 - x^5 + 5x^3 - 4x^2 - 5 = 5x^3 - 4x^2 - 5.
   \]

4. **Repeat the Process:**

   Divide the new leading term \(5x^3\) by \(x^2\):
   
   \[
   5x^3 \div x^2 = 5x.
   \]

   So, the next term in the quotient is \(5x\).

5. **Multiply and Subtract Again:**

   Multiply the entire divisor by \(5x\):
   
   \[
   (5x^3 - 4x^2 - 5) - (5x(x^2 - 5)) = (5x^3 - 4x^2 - 5) - (5x^3 - 25x).
   \]

   Simplifying, we get:
   
   \[
   5x^3 - 5x^3 - 4x^2 + 25x - 5 = -4x^2 + 25x - 5.
   \]

6. **Divide Again:**

   Divide the new leading term \(-4x^2\) by \(x^2\):
   
   \[
   -4x^2 \div x^2 = -4.
   \]

   So, the next term in the quotient is \(-4\).

7. **Multiply and Subtract One More Time:**

   Multiply the entire divisor by \(-4\):
   
   \[
   (-4x^2 + 25x - 5) - (-4(x^2 - 5)) = (-4x^2 + 25x - 5) - (-4x^2 + 20).
   \]

   Simplifying, we get:
   
   \[
   -4x^2 + 4x^2 + 25x - 5 - 20 = 25x - 25.
   \]

8. **Remainder:**

   The expression \(25x - 25\) is the remainder since its degree is less than the degree of the divisor \(x^2 - 5\).

### Final Answer:

\[
\frac{x^5 - 4x^2 - 5}{x^2 - 5} = x^3 + 5x - 4 + \frac{25x - 25}{x^2 - 5}.
\]

The quotient is \(x^3 + 5x - 4\) and the remainder is \(25x - 25\).

Would you like to see more details or have any questions?

### Related Questions:

1. What is polynomial long division, and why is it useful?
2. How do you find the remainder when dividing polynomials?
3. What are the steps to perform synthetic division on a polynomial?
4. How can the Remainder Theorem be applied in polynomial division?
5. What is the difference between polynomial long division and synthetic division?

### Tip:

Always align terms by their degrees when performing polynomial long division to avoid errors in subtraction.

Explore a detailed step-by-step guide on how to divide the polynomial x^5 - 4x^2 - 5 by x^2 - 5 using polynomial long division. This problem introduces key algebraic concepts, including polynomial division and the Polynomial Long Division Theorem. Ideal for students in Grades 10-12.

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