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