To find the missing side of the shape given the perimeter and the lengths of the other three sides, we can use the perimeter formula:
\[ \text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Missing Side} \]
We know:
- Perimeter = \( 21x^2 - 5x + 4 \)
- Side 1 = \( 2x - 3 \)
- Side 2 = \( 2x - 3 \)
- Side 3 = \( 7x^2 + 3x + 2 \)
Let's denote the length of the missing side as \( \text{Missing Side} = M \).
First, calculate the total length of sides 1, 2, and 3:
\[ \text{Total length of known sides} = (2x - 3) + (2x - 3) + (7x^2 + 3x + 2 \]
Combining the known sides:
\[ = (2x + 2x + 3x + 7x^2) + (-3 - 3 + 2) \]
Combining like terms:
\[ = 7x^2 + 7x - 4 \]
Now we set up the equation using the perimeter:
\[ 21x^2 - 5x + 4 = (7x^2 + 7x - 4) + M \]
To isolate \( M \), we rearrange the equation:
\[ M = (21x^2 - 5x + 4) - (7x^2 + 7x - 4) \]
Subtracting the known sides from the perimeter:
\[ M = 21x^2 - 5x + 4 - 7x^2 - 7x + 4 \]
Combine like terms:
\[ M = (21x^2 - 7x^2) + (-5x - 7x) + (4 + 4) \]
\[ M = 14x^2 - 12x + 8 \]
So, the missing side is:
\[ \boxed{14x^2 - 12x + 8} \]