To find the area of a rectangle, you can use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
In this case, the length is given by the polynomial \(x^2 + 2x + 4\) and the width is given by \(5x - 3\).
To find the area, we will multiply these two polynomials:
\[ \text{Area} = (x^2 + 2x + 4)(5x - 3) \]
Now we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:
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Multiply \(x^2\) by \(5x\) and \(-3\): \[ x^2 \cdot 5x = 5x^3 \] \[ x^2 \cdot (-3) = -3x^2 \]
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Multiply \(2x\) by \(5x\) and \(-3\): \[ 2x \cdot 5x = 10x^2 \] \[ 2x \cdot (-3) = -6x \]
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Multiply \(4\) by \(5x\) and \(-3\): \[ 4 \cdot 5x = 20x \] \[ 4 \cdot (-3) = -12 \]
Now we can combine all these results:
\[ \text{Area} = 5x^3 + (-3x^2 + 10x^2) + (-6x + 20x) - 12 \] \[ = 5x^3 + 7x^2 + 14x - 12 \]
Thus, the polynomial expression that represents the area of the stage is:
\[ \boxed{5x^3 + 7x^2 + 14x - 12} \]
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