To expand the expression \((2x^2 + x + 3)(3x^2 - 2x + 6)\) and write it in standard form, we will use the distributive property (also known as the FOIL method for binomials, but here we apply it to polynomials).
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Distribute each term in the first polynomial to every term in the second polynomial:
- \(2x^2 \cdot 3x^2\)
- \(2x^2 \cdot (-2x)\)
- \(2x^2 \cdot 6\)
- \(x \cdot 3x^2\)
- \(x \cdot (-2x)\)
- \(x \cdot 6\)
- \(3 \cdot 3x^2\)
- \(3 \cdot (-2x)\)
- \(3 \cdot 6\)
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Calculate each of these products:
- \(2x^2 \cdot 3x^2 = 6x^4\)
- \(2x^2 \cdot (-2x) = -4x^3\)
- \(2x^2 \cdot 6 = 12x^2\)
- \(x \cdot 3x^2 = 3x^3\)
- \(x \cdot (-2x) = -2x^2\)
- \(x \cdot 6 = 6x\)
- \(3 \cdot 3x^2 = 9x^2\)
- \(3 \cdot (-2x) = -6x\)
- \(3 \cdot 6 = 18\)
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Combine the results:
- \(6x^4\)
- Combine like terms for \(x^3\): \(-4x^3 + 3x^3 = -x^3\)
- Combine like terms for \(x^2\): \(12x^2 - 2x^2 + 9x^2 = 19x^2\)
- Combine like terms for \(x\): \(6x - 6x = 0\)
- The constant term is \(18\)
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Write the final polynomial in standard form: \[ \boxed{6x^4 - x^3 + 19x^2 + 18} \]
To clarify, the final answer is indeed a polynomial in standard form (which arranges the terms in descending order of their degrees) without parentheses around it, as standard form typically does not include parentheses.