To find the length of the missing side of the shape given that the perimeter is \(21x^2 - 5x + 4\), we first need to determine the sum of the known sides:
- Side 1: \(2x - 3\)
- Side 2: \(7x^2 + 3 + 2 = 7x^2 + 5\)
- Side 3: \(2x - 3\)
Now, let's find the combined length of these three sides:
\[ \text{Total length of known sides} = (2x - 3) + (7x^2 + 5) + (2x - 3) \]
Combining like terms:
\[ = 7x^2 + (2x + 2x) + (-3 - 3 + 5) \] \[ = 7x^2 + 4x - 1 \]
Now, let \(s\) be the length of the missing side. The equation for the perimeter is given by the sum of all sides:
\[ \text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Missing side} \] \[ 21x^2 - 5x + 4 = (7x^2 + 4x - 1) + s \]
To find \(s\), we can rearrange this equation:
\[ s = (21x^2 - 5x + 4) - (7x^2 + 4x - 1) \]
Now, perform the subtraction:
\[ s = 21x^2 - 5x + 4 - 7x^2 - 4x + 1 \] \[ = (21x^2 - 7x^2) + (-5x - 4x) + (4 + 1) \] \[ = 14x^2 - 9x + 5 \]
Thus, the length of the missing side is:
\[ \boxed{14x^2 - 9x + 5} \]
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