Long Division of Polynomials: (x^3 + 3x^2 + 3x + 2) / (x  1)
Long division is a fundamental technique in algebra for dividing polynomials. Let's illustrate this process with the example of dividing (x^3 + 3x^2 + 3x + 2) by (x  1).
Steps for Long Division

Set up the division: Write the dividend (x^3 + 3x^2 + 3x + 2) inside the division symbol and the divisor (x  1) outside.
_________ x  1  x^3 + 3x^2 + 3x + 2

Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2. Write this term above the division symbol, aligning it with the x^2 term in the dividend.
x^2 x  1  x^3 + 3x^2 + 3x + 2

Multiply and subtract: Multiply the quotient term (x^2) by the divisor (x  1), which gives us x^3  x^2. Write this result below the dividend, aligning the terms. Subtract this result from the dividend.
x^2 x  1  x^3 + 3x^2 + 3x + 2 x^3  x^2  4x^2 + 3x

Bring down the next term: Bring down the next term from the dividend (3x).
x^2 x  1  x^3 + 3x^2 + 3x + 2 x^3  x^2  4x^2 + 3x

Repeat the process: Repeat steps 24 with the new dividend (4x^2 + 3x). Divide the leading term (4x^2) by the leading term of the divisor (x), resulting in 4x. Multiply 4x by the divisor (x  1), which gives us 4x^2  4x. Subtract this from the current dividend.
x^2 + 4x x  1  x^3 + 3x^2 + 3x + 2 x^3  x^2  4x^2 + 3x 4x^2  4x  7x + 2

Bring down the next term: Bring down the next term from the dividend (2).
x^2 + 4x x  1  x^3 + 3x^2 + 3x + 2 x^3  x^2  4x^2 + 3x 4x^2  4x  7x + 2

Repeat the process: Divide the leading term of the current dividend (7x) by the leading term of the divisor (x), resulting in 7. Multiply 7 by the divisor (x  1), which gives us 7x  7. Subtract this from the current dividend.
x^2 + 4x + 7 x  1  x^3 + 3x^2 + 3x + 2 x^3  x^2  4x^2 + 3x 4x^2  4x  7x + 2 7x  7  9

The remainder: The final result is the quotient (x^2 + 4x + 7) and a remainder of 9.
Result
Therefore, (x^3 + 3x^2 + 3x + 2) / (x  1) = x^2 + 4x + 7 + 9/(x  1).