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The division problem you've provided is the polynomial division of:

\[
\frac{8x^4 + 6x^3 - 23x^2 + 15x^2 + 4}{4x^2 - 3x + 1}
\]

This is a long division of polynomials. Let me calculate the quotient and the remainder for this expression.

I'll go step by step:

### Step 1: Divide the first term of the numerator by the first term of the denominator:
\[
\frac{8x^4}{4x^2} = 2x^2
\]

### Step 2: Multiply the entire denominator by \(2x^2\):
\[
(4x^2 - 3x + 1) \times 2x^2 = 8x^4 - 6x^3 + 2x^2
\]

### Step 3: Subtract this from the original polynomial:
\[
(8x^4 + 6x^3 - 23x^2 + 15x + 4) - (8x^4 - 6x^3 + 2x^2) = 12x^3 - 25x^2 + 15x + 4
\]

### Step 4: Divide the first term of the new polynomial by the first term of the denominator:
\[
\frac{12x^3}{4x^2} = 3x
\]

### Step 5: Multiply the entire denominator by \(3x\):
\[
(4x^2 - 3x + 1) \times 3x = 12x^3 - 9x^2 + 3x
\]

### Step 6: Subtract this from the new polynomial:
\[
(12x^3 - 25x^2 + 15x + 4) - (12x^3 - 9x^2 + 3x) = -16x^2 + 12x + 4
\]

### Step 7: Divide the first term of the new polynomial by the first term of the denominator:
\[
\frac{-16x^2}{4x^2} = -4
\]

### Step 8: Multiply the entire denominator by \(-4\):
\[
(4x^2 - 3x + 1) \times (-4) = -16x^2 + 12x - 4
\]

### Step 9: Subtract this from the new polynomial:
\[
(-16x^2 + 12x + 4) - (-16x^2 + 12x - 4) = 8
\]

Thus, the quotient is:
\[
2x^2 + 3x - 4
\]

And the remainder is:
\[
8
\]

### Final Answer:
- Quotient: \(2x^2 + 3x - 4\)
- Remainder: \(8\)

---

Would you like me to explain any steps in more detail or clarify any part of the process?

Here are some related questions you might find helpful:
1. How do you check if the division was done correctly?
2. Can polynomial division be applied to higher-degree polynomials?
3. How would this process differ with complex numbers in the coefficients?
4. What are the applications of polynomial division in algebra?
5. How does synthetic division simplify polynomial division?

**Tip:** Polynomial division is similar to long division of numbers—focus on dividing the leading terms, multiplying, and subtracting systematically.

Indicar el cociente y el residuo de la división: (8x^4 + 6x^3 - 23x^2 + 15x^2 + 4) / (4x^2 - 3x + 1)

Find the quotient and remainder of the polynomial division (8x^4 + 6x^3 - 23x^2 + 15x^2 + 4) divided by (4x^2 - 3x + 1) using long division. Suitable for Grades 10-12 students learning advanced algebra concepts.

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