To find the product of the polynomials \((6x^2 + 4x - 3)(-3x + 2)\) using the area model, we can break this down into simpler parts.
Step 1: Set Up the Area Model
We will create a grid that corresponds to the terms in each polynomial. The first polynomial has three terms, and the second polynomial has two terms:
- For \(6x^2 + 4x - 3\): 1st row will be \(6x^2\), 2nd row will be \(4x\), and 3rd row will be \(-3\).
- For \(-3x + 2\): first column will be \(-3x\) and the second column will be \(2\).
This gives us a 3x2 grid:
\[ \begin{array}{c|c|c} & -3x & 2 \ \hline 6x^2 & 6x^2 \cdot (-3x) & 6x^2 \cdot 2 \ \hline 4x & 4x \cdot (-3x) & 4x \cdot 2 \ \hline -3 & -3 \cdot (-3x) & -3 \cdot 2 \ \end{array} \]
Step 2: Calculate Each Product
Now, we'll fill in the products for each cell in the grid.
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For the cell \(6x^2 \cdot (-3x)\): \[ 6x^2 \cdot (-3x) = -18x^3 \]
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For the cell \(6x^2 \cdot 2\): \[ 6x^2 \cdot 2 = 12x^2 \]
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For the cell \(4x \cdot (-3x)\): \[ 4x \cdot (-3x) = -12x^2 \]
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For the cell \(4x \cdot 2\): \[ 4x \cdot 2 = 8x \]
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For the cell \(-3 \cdot (-3x)\): \[ -3 \cdot (-3x) = 9x \]
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For the cell \(-3 \cdot 2\): \[ -3 \cdot 2 = -6 \]
Step 3: Combine the Products
Now we can combine all the products:
\[ -18x^3 + 12x^2 - 12x^2 + 8x + 9x - 6 \]
Step 4: Simplify
Now, we will simplify this expression by combining like terms:
- The \(x^2\) terms: \(12x^2 - 12x^2 = 0\)
- The \(x\) terms: \(8x + 9x = 17x\)
So the expression simplifies to:
\[ -18x^3 + 17x - 6 \]
Final Result
The product of the polynomials \((6x^2 + 4x - 3)(-3x + 2)\) in standard form is:
\[ \boxed{-18x^3 + 17x - 6} \]