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The problem shown in the image involves dividing the polynomial \(-10x^2 + 4x^3 + 3\) by \(x^2 - 2x + 1\). The goal is to express the result in the form of:

\[
\text{Quotient} + \frac{\text{Remainder}}{x^2 - 2x + 1}
\]

I will walk through the polynomial long division.

### Step 1: Arrange the polynomials in standard form
We need to divide:

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

### Step 2: Perform the division
- First, divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x^2\), which gives \(4x\).
- Multiply \(4x\) by the entire divisor \((x^2 - 2x + 1)\), which results in:

  \[
  4x \cdot (x^2 - 2x + 1) = 4x^3 - 8x^2 + 4x
  \]

- Subtract this from the original dividend:

  \[
  (4x^3 - 10x^2 + 3) - (4x^3 - 8x^2 + 4x) = -2x^2 - 4x + 3
  \]

### Step 3: Repeat the process
- Divide the new leading term \(-2x^2\) by \(x^2\), which gives \(-2\).
- Multiply \(-2\) by the divisor \((x^2 - 2x + 1)\), which gives:

  \[
  -2 \cdot (x^2 - 2x + 1) = -2x^2 + 4x - 2
  \]

- Subtract this from \(-2x^2 - 4x + 3\):

  \[
  (-2x^2 - 4x + 3) - (-2x^2 + 4x - 2) = -8x + 5
  \]

### Step 4: Write the result
The quotient is \(4x - 2\), and the remainder is \(-8x + 5\). Therefore, the division can be

Divide (-10x^2 + 4x^3 + 3) ÷ (x^2 - 2x + 1).

Learn how to divide the polynomial (-10x^2 + 4x^3 + 3) by (x^2 - 2x + 1) using long division. This example covers key algebraic concepts, polynomial division, and includes a detailed step-by-step solution suitable for high school students (Grades 10-12).

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