Explanation:
To solve this problem, we need to use the division algorithm for polynomials. According to the division algorithm, if we divide a polynomial f(x) by another polynomial g(x), we get a quotient q(x) and a remainder r(x) such that:

f(x) = g(x)q(x) + r(x)

In this problem, we are given that:

f(x) = x^3 - 3x^2 + 2x
q(x) = x - 2
r(x) = -2x - 4

We need to find g(x).

Step 1: Write the division equation using the given information.

x^3 - 3x^2 + 2x = g(x)(x - 2) - 2x - 4

Step 2: Simplify the equation by multiplying the divisor (x - 2) with the quotient g(x).

x^3 - 3x^2 + 2x = g(x)x - 2g(x) - 2x - 4

Step 3: Collect like terms and simplify.

x^3 - 3x^2 + 2x = (g(x) - 2)x - 2g(x) - 4

Step 4: Equate the coefficients of x^3, x^2, and x on both sides of the equation.

Coefficient of x^3: 1 = g(x)
Coefficient of x^2: -3 = 0 (there is no x^2 term on the right-hand side)
Coefficient of x: 2 = g(x) - 2

Solving for g(x), we get:

g(x) = 1
g(x) - 2 = 2
g(x) = 4

Step 5: Check the solution by substituting g(x) = 4 in the division equation.

x^3 - 3x^2 + 2x = 4(x - 2) - 2x - 4

Simplifying, we get:

x^3 - 3x^2 + 2x = 4x - 8 - 2x - 4
x^3 - 3x^2 + 2x = 2x - 12

This is true, so g(x) = 4 is the correct answer.

Final Answer: The polynomial g(x) is 4.