To expand the expression \((x^2 - 3x + 1)(x^2 + 4x + 4)\), we will use the distributive property (also known as the FOIL method for binomials) for each term in the first polynomial multiplied by each term in the second polynomial:
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First, distribute \(x^2\): \[ x^2 \cdot (x^2 + 4x + 4) = x^4 + 4x^3 + 4x^2 \]
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Next, distribute \(-3x\): \[ -3x \cdot (x^2 + 4x + 4) = -3x^3 - 12x^2 - 12x \]
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Finally, distribute \(1\): \[ 1 \cdot (x^2 + 4x + 4) = x^2 + 4x + 4 \]
Now, we combine all of these results together: \[ x^4 + 4x^3 + 4x^2 - 3x^3 - 12x^2 - 12x + x^2 + 4x + 4 \]
Next, we simplify by combining like terms:
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For \(x^4\): \[ x^4 \quad (\text{only one term}) \]
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For \(x^3\): \[ 4x^3 - 3x^3 = x^3 \]
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For \(x^2\): \[ 4x^2 - 12x^2 + x^2 = -7x^2 \]
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For \(x\): \[ -12x + 4x = -8x \]
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For the constant term: \[ 4 \quad (\text{only one term}) \]
Putting all these together, we get the expanded polynomial in standard form: \[ \boxed{x^4 + x^3 - 7x^2 - 8x + 4} \]