If \(x = 4 + i\), where \(i = \sqrt{-1}\), then what is \(x^3 + 2x^2 - 63x + 171\) equal to?
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Since \(x = 4 + i\), we have \((x-4)^2 = i^2 = -1\), giving \(x^2 - 8x + 17 = 0\). Dividing \(x^3 + 2x^2 - 63x + 171\) by \(x^2 - 8x + 17\) gives quotient \((x+10)\) and remainder \(1\), so \(x^3 + 2x^2 - 63x + 171 = (x^2-8x+17)(x+10) + 1 = 0 + 1 = 1\).
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