Divide 10 x² + 41 x square - 4 x minus 15 by 5 x + 3 using long division method

Let's divide \(10x^3 + 41x^2 - 4x - 15\) by \(5x + 3\) using the long division method.

### Step 1: Set Up the Division

We set up the division as follows:

\[
\begin{array}{r|l}
5x + 3 & 10x^3 + 41x^2 - 4x - 15 \\
\end{array}
\]

### Step 2: Divide the First Term

- Divide the first term of the dividend \(10x^3\) by the first term of the divisor \(5x\):
  \[
  \frac{10x^3}{5x} = 2x^2
  \]
- This is the first term of the quotient.

### Step 3: Multiply and Subtract

- Multiply the entire divisor \(5x + 3\) by the first term of the quotient \(2x^2\):
  \[
  2x^2 \times (5x + 3) = 10x^3 + 6x^2
  \]
- Subtract this from the original polynomial:
  \[
  (10x^3 + 41x^2 - 4x - 15) - (10x^3 + 6x^2) = 35x^2 - 4x - 15
  \]

### Step 4: Repeat the Process

- Now, divide the first term of the new polynomial \(35x^2\) by the first term of the divisor \(5x\):
  \[
  \frac{35x^2}{5x} = 7x
  \]
- Multiply the entire divisor \(5x + 3\) by \(7x\):
  \[
  7x \times (5x + 3) = 35x^2 + 21x
  \]
- Subtract this from the polynomial:
  \[
  (35x^2 - 4x - 15) - (35x^2 + 21x) = -25x - 15
  \]

### Step 5: Final Step

- Divide the first term of the new polynomial \(-25x\) by the first term of the divisor \(5x\):
  \[
  \frac{-25x}{5x} = -5
  \]
- Multiply the entire divisor \(5x + 3\) by \(-5\):
  \[
  -5 \times (5x + 3) = -25x - 15
  \]
- Subtract this from the polynomial:
  \[
  (-25x - 15) - (-25x - 15) = 0
  \]

### Final Answer

The quotient is:
\[
2x^2 + 7x - 5
\]
And there is no remainder.

Would you like further explanation on any part of this process, or do you have any specific questions?

### Follow-up Questions:
1. Would you like to try another long division example with different polynomials?
2. Do you need help understanding how to handle remainders in polynomial division?
3. Are there any concepts in this division that you'd like clarified?
4. Would you like to compare this result with synthetic division for the same polynomials?
5. Do you want to explore more complex polynomials and division methods?

**Tip**: Always align terms correctly when performing polynomial long division, particularly when subtracting each step, to avoid errors.

Learn how to divide the polynomial 10x^3 + 41x^2 - 4x - 15 by 5x + 3 using the long division method. Follow a detailed step-by-step solution with explanations. Suitable for advanced high school students and those studying polynomial division.

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