To find the length of the missing side of the shape, we will first establish the perimeter formula and the lengths of the given sides.
Given that the shape has two sides measuring \(2x - 3\) each and one side measuring \(7x^2 + 3x + 2\), we can denote the missing side's length as \(y\).
The perimeter \(P\) of the shape can be calculated as: \[ P = \text{left side} + \text{right side} + \text{bottom side} + \text{missing side} \]
Substituting the known values, the equation for the perimeter is: \[ P = (2x - 3) + (2x - 3) + (7x^2 + 3x + 2) + y \] This simplifies to: \[ P = 2(2x - 3) + (7x^2 + 3x + 2) + y \] Expanding this gives: \[ P = 4x - 6 + 7x^2 + 3x + 2 + y \] Combining like terms results in: \[ P = 7x^2 + (4x + 3x) - 6 + 2 + y \] Which simplifies to: \[ P = 7x^2 + 7x - 4 + y \]
We know from the problem that the given perimeter is: \[ P = 21x^2 - 5x + 4 \]
Now, we can set the two expressions for the perimeter equal to each other: \[ 7x^2 + 7x - 4 + y = 21x^2 - 5x + 4 \]
To isolate \(y\), we can rearrange and simplify: \[ y = (21x^2 - 5x + 4) - (7x^2 + 7x - 4) \] Distributing the negative sign gives: \[ y = 21x^2 - 5x + 4 - 7x^2 - 7x + 4 \] Combining like terms results in: \[ y = (21x^2 - 7x^2) + (-5x - 7x) + (4 + 4) \] This simplifies to: \[ y = 14x^2 - 12x + 8 \]
Therefore, the length of the missing side is: \[ \boxed{14x^2 - 12x + 8} \]
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