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The problem you're working on is a polynomial division:

\[
\frac{10x^3 + 11x + 4}{5x^2 + 1}
\]

### Steps to Solve the Division:
We are dividing a cubic polynomial by a quadratic polynomial, so let's go through the process of polynomial long division.

1. **First Term of Quotient**: 
   Divide the leading term of the numerator \(10x^3\) by the leading term of the denominator \(5x^2\):
   \[
   \frac{10x^3}{5x^2} = 2x
   \]
   So, the first term of the quotient is \(2x\).

2. **Multiply and Subtract**:
   Now, multiply \(2x\) by the divisor \(5x^2 + 1\):
   \[
   2x(5x^2 + 1) = 10x^3 + 2x
   \]
   Subtract this from the original polynomial:
   \[
   (10x^3 + 11x + 4) - (10x^3 + 2x) = 9x + 4
   \]

3. **Next Term of Quotient**:
   Divide the leading term of the remaining polynomial \(9x\) by the leading term \(5x^2\). Since \(9x\) cannot be divided by \(5x^2\) (lower degree), this will be the remainder.

### Final Answer:
Thus, the quotient is \(2x\) and the remainder is \(9x + 4\). So, the division can be written as:

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

Would you like more details or further clarification?

Here are some related questions:
1. How do we handle polynomial division when the divisor is a higher degree?
2. What happens when the remainder is zero in polynomial division?
3. Can this division be done using synthetic division?
4. What are the real-world applications of polynomial division?
5. How does the degree of polynomials affect the quotient and remainder?

**Tip**: Always align terms by degree when subtracting during polynomial division to avoid errors.

Divide (10x^3 + 11x + 4) ÷ (5x^2 + 1) and write the answer in the form: Quotient + Remainder / (5x^2 + 1).

Learn how to perform polynomial division by dividing 10x^3 + 11x + 4 by 5x^2 + 1. This guide provides a step-by-step solution, including determining the quotient and remainder. Suitable for high school students in Grades 9-12.

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