To multiply the polynomials \((6x^2 + 4x - 3)(-3x + 2)\) using the area model, we can break it down as follows:
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Set up the area model:
- Create a grid. Each term of the first polynomial will label one side, and each term of the second polynomial will label the other side.
- The terms from the first polynomial \(6x^2\), \(4x\), and \(-3\) will be placed on one side, and the terms from the second polynomial \(-3x\) and \(2\) will be placed on the other side.
Here’s how it looks:
\[ \begin{array}{c|c|c} & -3x & 2 \ \hline 6x^2 & 6x^2 \cdot (-3x) & 6x^2 \cdot 2 \ \hline 4x & 4x \cdot (-3x) & 4x \cdot 2 \ \hline -3 & -3 \cdot (-3x) & -3 \cdot 2 \ \end{array} \]
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Calculate each area in the grid:
- For \(6x^2 \cdot -3x\): \[ 6x^2 \cdot -3x = -18x^3 \]
- For \(6x^2 \cdot 2\): \[ 6x^2 \cdot 2 = 12x^2 \]
- For \(4x \cdot -3x\): \[ 4x \cdot -3x = -12x^2 \]
- For \(4x \cdot 2\): \[ 4x \cdot 2 = 8x \]
- For \(-3 \cdot -3x\): \[ -3 \cdot -3x = 9x \]
- For \(-3 \cdot 2\): \[ -3 \cdot 2 = -6 \]
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Combine all the results together:
Now, we add all these products together: \[ -18x^3 + 12x^2 - 12x^2 + 8x + 9x - 6 \]
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Combine like terms:
- The \(x^2\) terms: \[ 12x^2 - 12x^2 = 0 \]
- The \(x\) terms: \[ 8x + 9x = 17x \]
So we have: \[ -18x^3 + 0 + 17x - 6 \]
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Final result in standard form:
Thus, the simplified product of the polynomials in standard form is: \[ \boxed{-18x^3 + 17x - 6} \]