To find the product of the two polynomials \( (2x^2 + x + 3)(x - 2) \), we can use the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first polynomial by each term in the second polynomial.
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Multiply \( 2x^2 \) by \( x \): \[ 2x^2 \cdot x = 2x^3 \]
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Multiply \( 2x^2 \) by \( -2 \): \[ 2x^2 \cdot (-2) = -4x^2 \]
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Multiply \( x \) by \( x \): \[ x \cdot x = x^2 \]
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Multiply \( x \) by \( -2 \): \[ x \cdot (-2) = -2x \]
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Multiply \( 3 \) by \( x \): \[ 3 \cdot x = 3x \]
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Multiply \( 3 \) by \( -2 \): \[ 3 \cdot (-2) = -6 \]
Now, we combine all of these results: \[ 2x^3 - 4x^2 + x^2 - 2x + 3x - 6 \]
Next, we combine like terms:
- The \( x^2 \) terms: \( -4x^2 + x^2 = -3x^2 \)
- The \( x \) terms: \( -2x + 3x = x \)
Putting it all together, we have: \[ 2x^3 - 3x^2 + x - 6 \]
Therefore, the product of \( (2x^2 + x + 3)(x - 2) \) is: \[ \boxed{2x^3 - 3x^2 + x - 6} \]