Find the GCD of x³ + 4x² + x − 6 and x³ − 3x² − 6x + 8 by the division method.

1. 1 · Write what is given and what is to be found. f(x) = x³ + 4x² + x − 6 and g(x) = x³ − 3x² − 6x + 8. To find: their GCD.
2. 2 · Write the method. Use the division method: divide, then divide the divisor by the remainder, until the remainder is zero.
3. 3 · First division. f(x) ÷ g(x): quotient 1, remainder 7x² + 7x − 14 = 7(x² + x − 2). Drop the 7.
4. 4 · Second division. g(x) ÷ (x² + x − 2): quotient x − 4, remainder 0.
5. 5 · Check and box the answer. (x² + x − 2)(x − 4) = x³ − 3x² − 6x + 8 = g(x). Box: GCD = x² + x − 2.

Scoring tip: Show each long division in full, with the subtraction lines. Each correct remainder earns a step mark.