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The problem is asking to divide:

\[
(10x^4 - x^3 - 3x^2) \text{ by } (2x^2 + x)
\]

### Solution:

We will perform polynomial long division.

1. **Step 1: Divide the leading term of the dividend by the leading term of the divisor**.
   
   \[
   \frac{10x^4}{2x^2} = 5x^2
   \]
   
   Now, multiply \(5x^2\) by the divisor \(2x^2 + x\):
   
   \[
   5x^2 \cdot (2x^2 + x) = 10x^4 + 5x^3
   \]
   
2. **Step 2: Subtract this from the original polynomial**:
   
   \[
   (10x^4 - x^3 - 3x^2) - (10x^4 + 5x^3) = -6x^3 - 3x^2
   \]
   
3. **Step 3: Divide the new leading term by the leading term of the divisor**:
   
   \[
   \frac{-6x^3}{2x^2} = -3x
   \]
   
   Multiply \(-3x\) by the divisor \(2x^2 + x\):
   
   \[
   -3x \cdot (2x^2 + x) = -6x^3 - 3x^2
   \]
   
4. **Step 4: Subtract again**:
   
   \[
   (-6x^3 - 3x^2) - (-6x^3 - 3x^2) = 0
   \]

Thus, the quotient is:

\[
5x^2 - 3x
\]

There is no remainder.

### Final Answer:
\[
\frac{10x^4 - x^3 - 3x^2}{2x^2 + x} = 5x^2 - 3x
\]

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How would the process change if the divisor was \(2x^2 + x + 1\)?
2. Can you perform synthetic division on this polynomial division?
3. What is the remainder if you divide \(10x^4 - x^3 - 3x^2 + 1\) by \(2x^2 + x\)?
4. How can you factor the expression \(10x^4 - x^3 - 3x^2\)?
5. What happens if the degree of the divisor is higher than the dividend?

### Tip:
Always align terms by powers of \(x\) while performing polynomial long division to avoid errors.

Divide (10x^4 - x^3 - 3x^2) by (2x^2 + x)

Learn how to divide the polynomial (10x^4 - x^3 - 3x^2) by (2x^2 + x) using polynomial long division. This problem covers key algebraic concepts and steps for polynomial division, suitable for students in Grades 10-12.

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