In order to find the length of the missing side of the shape given the perimeter \( P = 21x^2 - 5x + 4 \), we first need to express the perimeter in terms of its sides.
Assume the shape is a polygon with \( 5 \) sides, where we know the lengths of \( 4 \) sides and we denote the missing side as \( S \). The perimeter will be the sum of all the sides:
\[ P = S + s_1 + s_2 + s_3 + s_4 \]
If \( s_1, s_2, s_3, \) and \( s_4 \) are known, we can rearrange this to solve for the missing side:
\[ S = P - (s_1 + s_2 + s_3 + s_4) \]
Without additional context or knowing the values of \( s_1, s_2, s_3, \) and \( s_4 \), it is impossible to find the length of the missing side directly. However, since we are also given options, we can assume one of these options should equal \( S \) if \( s_1, s_2, s_3, \) and \( s_4 \) are calculated.
To figure out which of the given options could be the missing side, we need to look at what \( S \) would equal when combined with the \( 4 \) sides. Since that information is not provided, we can check each option's ability to combine with constants to match the provided perimeter.
Given the options:
A) \( 4x^2 + 5 \)
B) \( -14x^2 + 12x - 8 \)
C) \( 14x^2 - 12x + 8 \)
D) \( 4x^2 - 8 \)
To check which of these options might make sense, we can't definitively solve for missing side length without additional context about the other sides. Since we don’t have values for \( s_1, s_2, s_3, \) and \( s_4 \), I would presuppose with various configurations to determine what leads to the right perimeter.
If we assume one of these sides could logically fit together with polynomials, for example, we could try subtracting permutations of them (pick one and check against the perimeter).
If you know any specifics about the other side lengths or the configuration, that would narrow it down significantly.
Without further information, in terms of polynomial degrees or coefficients matching, option C) \( 14x^2 - 12x + 8 \) is of a similar type of polynomial compared to the perimeter equation and may imply a structural balance, as \( 21x^2 - 5x + 4 \) includes a positive degree leading it up, congruently matching with the values to help complete the perimeter.
In summary, without the values for \( s_1, s_2, s_3, \) and \( s_4 \), the rationale remains incomplete for an easier path.