The sum of the coefficients of \(x^4\) and \(x^2\) in the product of \((x^2 - x + 4)\) and \((x^3 - 2x^2 + 3x + 1)\) is:
-13
Multiplying term by term, \(x^2(x^3-2x^2+3x+1) = x^5 - 2x^4 + 3x^3 + x^2\).
Next, \(-x(x^3-2x^2+3x+1) = -x^4 + 2x^3 - 3x^2 - x\).
Next, \(4(x^3-2x^2+3x+1) = 4x^3 - 8x^2 + 12x + 4\).
Adding all three results, the \(x^4\) terms give \(-2x^4 - x^4 = -3x^4\), and the \(x^2\) terms give \(x^2 - 3x^2 - 8x^2 = -10x^2\).
So the coefficient of \(x^4\) is \(-3\) and the coefficient of \(x^2\) is \(-10\), and their sum is \(-3 + (-10) = -13\).
Hence, the sum of the coefficients of \(x^4\) and \(x^2\) is \(-13\).
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