To find the polynomial expression that represents the perimeter of the rectangular poster board, we use the formula for the perimeter \( P \) of a rectangle, which is given by:
\[ P = 2(\text{length} + \text{width}) \]
In this case, the length \( L \) is given as:
\[ L = x^2 - x + 1 \]
And the width \( W \) is given as:
\[ W = 2x^2 + 3x - 2 \]
First, we will add the length and width together:
\[ L + W = (x^2 - x + 1) + (2x^2 + 3x - 2) \]
Now combine like terms:
\[ L + W = x^2 + 2x^2 - x + 3x + 1 - 2 \] \[ = (1 + 2)x^2 + (-1 + 3)x + (1 - 2) \] \[ = 3x^2 + 2x - 1 \]
Next, we substitute this expression back into the perimeter formula:
\[ P = 2(L + W) = 2(3x^2 + 2x - 1) \]
Now, distribute the 2:
\[ P = 2 \cdot 3x^2 + 2 \cdot 2x + 2 \cdot (-1) \] \[ = 6x^2 + 4x - 2 \]
Thus, the polynomial expression that represents the perimeter of the poster board is:
\[ \boxed{6x^2 + 4x - 2} \]
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