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x^4 + 3x^3 - 2x^2 + x + 5 = (x^2 - x + 1)(x^2 + ax + b) + ax + c
Expand right-hand side:
= x^4 + (a - 1)x^3 + (b - a + 1)x^2 + (-b + a)x + b
Add \(ax + c\):
x^4 + (a - 1)x^3 + (b - a + 1)x^2 + (-b + a + a)x + (b + c) = x^4 + (a - 1)x^3 + (b - a + 1)x^2 + (2a - b)x + (b + c)
Match coefficients:
\(x^4\): 1 = 1 â
\(x^3\): \(a - 1 = 3 \Rightarrow a = 4\)
\(x^2\): \(b - a + 1 = -2\), \(b - 4 + 1 = -2 \Rightarrow b = 1\)
\(x\): \(2a - b = 1 \Rightarrow 8 - 1 = 7