To find the product of the polynomials \((2x + 1)(x - 4)\), we can use the distributive property (also known as the FOIL method for binomials).
Here are the steps to multiply the two polynomials:
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Distribute \(2x\) to both terms in the second polynomial: \[ 2x \cdot x + 2x \cdot (-4) = 2x^2 - 8x \]
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Distribute \(1\) to both terms in the second polynomial: \[ 1 \cdot x + 1 \cdot (-4) = x - 4 \]
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Now, combine all the terms from steps 1 and 2: \[ 2x^2 - 8x + x - 4 \]
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Combine like terms: \[ 2x^2 - 7x - 4 \]
Thus, the product of the polynomials \((2x + 1)(x - 4)\) is: \[ \boxed{2x^2 - 7x - 4} \]