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The problem you're working on is polynomial long division. The goal is to divide the polynomial 

\[
-6x^4 + 2x^3 + 25x^2 - 12x + 25
\]

by

\[
x^2 - 5
\]

and express the result in the form:

\[
q(x) + \frac{r(x)}{x^2 - 5}
\]

where \( q(x) \) is the quotient and \( r(x) \) is the remainder.

### Steps:
1. **Divide the first term of the numerator by the first term of the denominator**: \(-6x^4 \div x^2 = -6x^2\). This is the first term of \( q(x) \).
   
2. **Multiply** \(-6x^2\) by the denominator \(x^2 - 5\) to get \(-6x^4 + 30x^2\).

3. **Subtract** this from the original polynomial:

\[
(-6x^4 + 2x^3 + 25x^2 - 12x + 25) - (-6x^4 + 30x^2) = 2x^3 - 5x^2 - 12x + 25
\]

4. **Repeat** the process:
   - Divide \(2x^3\) by \(x^2\), which gives \(2x\).
   - Multiply \(2x\) by \(x^2 - 5\) to get \(2x^3 - 10x\).
   - Subtract: 

\[
(2x^3 - 5x^2 - 12x + 25) - (2x^3 - 10x) = -5x^2 - 2x + 25
\]

5. **Divide** \(-5x^2\) by \(x^2\), which gives \(-5\).
   - Multiply \(-5\) by \(x^2 - 5\) to get \(-5x^2 + 25\).
   - Subtract:

\[
(-5x^2 - 2x + 25) - (-5x^2 + 25) = -2x
\]

At this point, there are no more terms in the numerator with degree 2 or higher. So, the quotient is:

\[
q(x) = -6x^2 + 2x - 5
\]

and the remainder is:

\[
r(x) = -2x
\]

Thus, the division is:

\[
\frac{-6x^4 + 2x^3 + 25x^2 - 12x + 25}{x^2 - 5} = (-6x^2 + 2x - 5) + \frac{-2x}{x^2 - 5}
\]

Would you like further details on any of these steps? 

Here are five related questions to explore:
1. How can you perform synthetic division instead of long division?
2. What happens if the degree of the denominator is greater than the numerator?
3. How do you verify the result of polynomial division?
4. What are common errors in polynomial long division?
5. Can you perform polynomial division when the denominator is not a quadratic?

**Tip:** Always align terms by degree when performing polynomial division to avoid confusion in subtraction.

Rewrite \((-6x^4 + 2x^3 + 25x^2 - 12x + 25) \div (x^2 - 5)\) in the form \( q(x) + \frac{r(x)}{x^2 - 5} \).

This problem demonstrates polynomial long division by dividing \( (-6x^4 + 2x^3 + 25x^2 - 12x + 25) \) by \( x^2 - 5 \). The solution includes finding the quotient and remainder, following step-by-step long division techniques suitable for high school students in Grades 10-12.

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