Write the quotient and remainder when we divide :
`(x^(4) - 2x^(3) + 2x^(2) + x + 4)` by `(x^(2) + x + 1)`

Answer

To solve the problem of dividing the polynomial \( x^4 - 2x^3 + 2x^2 + x + 4 \) by \( x^2 + x + 1 \), we will use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \( x^4 - 2x^3 + 2x^2 + x + 4 \) by \( x^2 + x + 1 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( x^4 \) by the leading term of the divisor \( x^2 \):
\[ \frac{x^4}{x^2} = x^2 \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Multiply the entire divisor \( x^2 + x + 1 \) by \( x^2 \):
\[ x^2(x^2 + x + 1) = x^4 + x^3 + x^2 \]
Now subtract this from the original polynomial:
\[ (x^4 - 2x^3 + 2x^2 + x + 4) - (x^4 + x^3 + x^2) = -3x^3 + x^2 + x + 4 \]

### Step 4: Repeat the process
Now, take the new polynomial \( -3x^3 + x^2 + x + 4 \) and repeat the process. Divide the leading term \( -3x^3 \) by \( x^2 \):
\[ \frac{-3x^3}{x^2} = -3x \]
This is the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply the divisor by \( -3x \):
\[ -3x(x^2 + x + 1) = -3x^3 - 3x^2 - 3x \]
Subtract this from the polynomial:
\[ (-3x^3 + x^2 + x + 4) - (-3x^3 - 3x^2 - 3x) = 4x^2 + 4x + 4 \]

### Step 6: Continue the process
Now, take \( 4x^2 + 4x + 4 \) and divide \( 4x^2 \) by \( x^2 \):
\[ \frac{4x^2}{x^2} = 4 \]
This is the next term of the quotient.

### Step 7: Final multiplication and subtraction
Multiply the divisor by \( 4 \):
\[ 4(x^2 + x + 1) = 4x^2 + 4x + 4 \]
Subtract this from the polynomial:
\[ (4x^2 + 4x + 4) - (4x^2 + 4x + 4) = 0 \]

### Conclusion
Since we have reached a remainder of \( 0 \), we can conclude that the quotient is:
\[ \text{Quotient} = x^2 - 3x + 4 \]
and the remainder is:
\[ \text{Remainder} = 0 \]

### Summary
When dividing \( x^4 - 2x^3 + 2x^2 + x + 4 \) by \( x^2 + x + 1 \), the quotient is \( x^2 - 3x + 4 \) and the remainder is \( 0 \).