To find the length of the missing side of the shape, we can start by utilizing the formula for perimeter, which is the sum of all sides. Given the perimeter \( P \) and the lengths of the sides, we can represent this relationship as:
\[ P = \text{side 1} + \text{side 2} + \text{side 3} + \text{missing side} \]
From your information, we know:
- Perimeter \( P = 21x^2 - 5x + 4 \)
- Side 1 = \( 2x - 3 \)
- Side 2 = \( 7x^2 + 3x + 2 \)
- Side 3 = \( 2x - 3 \)
- Let's denote the missing side as \( S \).
Now, substituting the sides into the perimeter equation, we have:
\[ 21x^2 - 5x + 4 = (2x - 3) + (7x^2 + 3x + 2) + (2x - 3) + S \]
Now, simplifying the right-hand side step by step:
- Combine like terms:
- The constant terms: \( -3 + 2 - 3 = -4 \)
- The linear terms: \( 2x + 3x + 2x = 7x \)
- The quadratic terms remain the same: \( 7x^2 \)
So, the new equation becomes:
\[ 21x^2 - 5x + 4 = 7x^2 + 7x - 4 + S \]
- To isolate \( S \), we rearrange the equation:
\[ S = (21x^2 - 5x + 4) - (7x^2 + 7x - 4) \]
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Now, simplify the right-hand side:
- Quadratic terms: \( 21x^2 - 7x^2 = 14x^2 \)
- Linear terms: \( -5x - 7x = -12x \)
- Constant terms: \( 4 + 4 = 8 \)
Putting it all together, we have:
\[ S = 14x^2 - 12x + 8 \]
Thus, the length of the missing side is:
\[ \boxed{14x^2 - 12x + 8} \]