To find the length of the missing side given the perimeter of the shape and the lengths of the other sides, we need to first express the perimeter in terms of the lengths of all the sides.
Let's denote the sides of the shape as follows:
- Side 1: \(2x - 3\)
- Side 2: \(7x^2 + 3x + 2\)
- Side 3: \(2x - 3\)
- Side 4: \(L\) (the missing side)
The perimeter \(P\) is given by: \[ P = \text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Side 4} \]
Substituting in the values we have: \[ 21x^2 - 5x + 4 = (2x - 3) + (7x^2 + 3x + 2) + (2x - 3) + L \]
Combining the known sides (Side 1, Side 2, Side 3): \[ = (2x - 3) + (2x - 3) + (7x^2 + 3x + 2) = (2x + 2x + 3x) + (7x^2) + (-3 - 3 + 2) \] \[ = 7x^2 + 7x - 4 \]
Now, we set the equation for the perimeter: \[ 21x^2 - 5x + 4 = (7x^2 + 7x - 4) + L \]
To find \(L\), we rearrange the equation: \[ L = (21x^2 - 5x + 4) - (7x^2 + 7x - 4) \]
We can simplify further by distributing the negative sign: \[ L = 21x^2 - 5x + 4 - 7x^2 - 7x + 4 \]
Combining like terms: \[ = (21x^2 - 7x^2) + (-5x - 7x) + (4 + 4) \] \[ = 14x^2 - 12x + 8 \]
Thus, the length of the missing side is: \[ \boxed{14x^2 - 12x + 8} \]