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.