To find the length of the missing side of the shape with a perimeter of \( 21x^2 - 5x + 4 \), we need to know how many sides are involved and their dimensions.
Typically, in a geometric shape such as a rectangle or polygon, the perimeter is calculated as the sum of all the sides. Without loss of generality, let's assume that we already know the lengths of the other sides. Let's call the known side lengths \( A_1, A_2, A_3, \) and \( A_4 \), and let \( A_5 \) be the unknown side.
If we know the expressions for \( A_1, A_2, A_3, \) and \( A_4 \), we can set up the equation for the perimeter:
\[ A_1 + A_2 + A_3 + A_4 + A_5 = 21x^2 - 5x + 4 \]
To isolate \( A_5 \) (the missing side), we can rearrange this equation:
\[ A_5 = (21x^2 - 5x + 4) - (A_1 + A_2 + A_3 + A_4) \]
Without specific values for \( A_1, A_2, A_3, \) and \( A_4 \), we can't solve for \( A_5 \) directly. However, we can analyze the answer choices you've provided.
If we consider the given options:
- \( 4x^2 + 5 \)
- \( -14x^2 + 12x - 8 \)
- \( 14x^2 - 12x + 8 \)
- \( 4x^2 - 8 \)
The value of the missing side \( A_5 \) must be consistent with the overall expression of the perimeter \( 21x^2 - 5x + 4 \). You can evaluate the options by substituting values for \( x \) or directly using relative polynomial degrees and coefficients to determine which is valid, or to check against previous sums, if known.
Assuming you have calculated or know the sum of the other side lengths \( A_1 + A_2 + A_3 + A_4 \) to determine \( A_5 \), you can substitute back accordingly.
If you could provide values for the known sides, I can assist further in calculating \( A_5 \). Otherwise, based on the options alone, the corresponding calculations should lead toward identifying which polynomial expression correctly satisfies this perimeter relationship.