To expand the expression \((2x^2 + x + 3)(3x^2 - 2x + 6)\), we will use the distributive property (also known as the FOIL method for binomials, but here we will extend it to polynomials).
We will multiply each term in the first polynomial by each term in the second polynomial:
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Multiply \(2x^2\) by each term in \(3x^2 - 2x + 6\): \[ 2x^2 \cdot 3x^2 = 6x^4 \] \[ 2x^2 \cdot (-2x) = -4x^3 \] \[ 2x^2 \cdot 6 = 12x^2 \]
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Multiply \(x\) by each term in \(3x^2 - 2x + 6\): \[ x \cdot 3x^2 = 3x^3 \] \[ x \cdot (-2x) = -2x^2 \] \[ x \cdot 6 = 6x \]
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Multiply \(3\) by each term in \(3x^2 - 2x + 6\): \[ 3 \cdot 3x^2 = 9x^2 \] \[ 3 \cdot (-2x) = -6x \] \[ 3 \cdot 6 = 18 \]
Now, combine all the results: \[ 6x^4 + (-4x^3 + 3x^3) + (12x^2 - 2x^2 + 9x^2) + (6x - 6x) + 18 \]
This simplifies to: \[ 6x^4 + (-4x^3 + 3x^3) + (12x^2 - 2x^2 + 9x^2) + 18 \] \[ = 6x^4 - x^3 + 19x^2 + 18 \]
Thus, the polynomial in standard form is: \[ 6x^4 - x^3 + 19x^2 + 18 \]
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