$$  $$$$ \frac{\left(5x^4+10x^3+9x^2+1\right)}{\left(5x^2-1\right)} $$

You provided the expression:

\[
\frac{5x^4 + 10x^3 + 9x^2 + 1}{5x^2 - 1}
\]

Let's simplify this rational expression by performing polynomial division. 

### Step 1: Perform long division of the polynomials.

We'll divide $ 5x^4 + 10x^3 + 9x^2 + 1 $ by $ 5x^2 - 1 $.

1. Divide the leading term of the numerator $ 5x^4 $ by the leading term of the denominator $ 5x^2 $, which gives $ x^2 $.
2. Multiply $ x^2 $ by $ 5x^2 - 1 $, resulting in $ 5x^4 - x^2 $.
3. Subtract $ (5x^4 - x^2) $ from the numerator:  
   \[
   (5x^4 + 10x^3 + 9x^2 + 1) - (5x^4 - x^2) = 10x^3 + 10x^2 + 1
   \]

4. Divide the leading term $ 10x^3 $ by the leading term $ 5x^2 $, giving $ 2x $.
5. Multiply $ 2x $ by $ 5x^2 - 1 $, resulting in $ 10x^3 - 2x $.
6. Subtract $ (10x^3 - 2x) $ from $ 10x^3 + 10x^2 + 1 $:
   \[
   (10x^3 + 10x^2 + 1) - (10x^3 - 2x) = 10x^2 + 2x + 1
   \]

7. Divide the leading term $ 10x^2 $ by $ 5x^2 $, giving $ 2 $.
8. Multiply $ 2 $ by $ 5x^2 - 1 $, resulting in $ 10x^2 - 2 $.
9. Subtract $ (10x^2 - 2) $ from $ 10x^2 + 2x + 1 $:
   \[
   (10x^2 + 2x + 1) - (10x^2 - 2) = 2x + 3
   \]

So, the quotient is $ x^2 + 2x + 2 $, with a remainder of $ 2x + 3 $. 

Thus, we can express the result as:

\[
\frac{5x^4 + 10x^3 + 9x^2 + 1}{5x^2 - 1} = x^2 + 2x + 2 + \frac{2x + 3}{5x^2 - 1}
\]

Would you like more details on any part of this process?

### Follow-up Questions:
1. What is polynomial long division, and how does it apply in rational expressions?
2. How do you check if your polynomial division is correct?
3. What are other methods to simplify rational expressions besides division?
4. How would this expression change if the degree of the numerator was lower than the denominator?
5. Can polynomial division always be performed, or are there special cases?

**Tip:** Always check if a polynomial can be factored before dividing—it can sometimes simplify the division!

$$ \frac{\left(5x^4+10x^3+9x^2+1\right)}{\left(5x^2-1\right)} $$

Learn how to simplify the rational expression $ \frac{5x^4 + 10x^3 + 9x^2 + 1}{5x^2 - 1} $ using polynomial long division. This detailed solution explains each step, focusing on algebraic techniques for handling rational expressions. Suitable for students in Grades 10-12.

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