On dividing `(x^(3)-3x^(2)+x+2)` by a polynomial g(x), the quotient and remainder are (x-2) and (-2x+4) respectively. Find g(x).

Explanation

To find the polynomial \( g(x) \) given that dividing \( p(x) = x^3 - 3x^2 + x + 2 \) by \( g(x) \) yields a quotient \( q(x) = x - 2 \) and a remainder \( r(x) = -2x + 4 \), we can use the polynomial division relationship:

\[ p(x) = g(x) \cdot q(x) + r(x) \]

### Step 1: Write the equation using the given information

We know:
- \( p(x) = x^3 - 3x^2 + x + 2 \)
- \( q(x) = x - 2 \)
- \( r(x) = -2x + 4 \)

Substituting these into the equation gives us:

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

### Step 2: Rearrange to isolate \( g(x)(x - 2) \)

Rearranging the equation, we have:

\[ g(x)(x - 2) = p(x) - r(x) \]

Substituting \( p(x) \) and \( r(x) \):

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

### Step 3: Simplify the right-hand side

Now, simplify the right-hand side:

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

Combining like terms:

\[ g(x)(x - 2) = x^3 - 3x^2 + 3x - 2 \]

### Step 4: Divide both sides by \( (x - 2) \)

Now, to find \( g(x) \), we need to divide both sides by \( (x - 2) \):

\[ g(x) = \frac{x^3 - 3x^2 + 3x - 2}{x - 2} \]

### Step 5: Perform the polynomial long division

Now, we will perform polynomial long division of \( x^3 - 3x^2 + 3x - 2 \) by \( x - 2 \).

1. Divide the leading term: \( x^3 \div x = x^2 \)
2. Multiply \( x^2 \) by \( (x - 2) \): \( x^2(x - 2) = x^3 - 2x^2 \)
3. Subtract:
\[ (x^3 - 3x^2) - (x^3 - 2x^2) = -3x^2 + 2x^2 = -x^2 \]
4. Bring down the next term:
\[ -x^2 + 3x \]
5. Divide the leading term: \( -x^2 \div x = -x \)
6. Multiply: \( -x(x - 2) = -x^2 + 2x \)
7. Subtract:
\[ (-x^2 + 3x) - (-x^2 + 2x) = 3x - 2x = x \]
8. Bring down the last term:
\[ x - 2 \]
9. Divide the leading term: \( x \div x = 1 \)
10. Multiply: \( 1(x - 2) = x - 2 \)
11. Subtract:
\[ (x - 2) - (x - 2) = 0 \]

Thus, we have:

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

### Final Answer

The polynomial \( g(x) \) is:

\[ \boxed{x^2 - x + 1} \]