To find the length of the missing side of the shape, we need to use the information given about the perimeter. The perimeter \( P \) of a shape is the sum of the lengths of all its sides.
Let \( P = 21x^2 - 5x + 4 \) represent the perimeter and assume we have a total of four sides, but we know the lengths of three of them. Let's denote the lengths of these sides as follows:
- Side 1: \( 4x^2 + 5 \)
- Side 2: \( -14x^2 + 12x - 8 \)
- Side 3: \( 14x^2 - 12x - 8 \)
Let’s denote the missing side as \( S \). According to the perimeter formula:
\[ P = \text{Side 1} + \text{Side 2} + \text{Side 3} + S \]
Substituting in the expressions for the sides we have:
\[ 21x^2 - 5x + 4 = (4x^2 + 5) + (-14x^2 + 12x - 8) + (14x^2 - 12x - 8) + S \]
Let’s simplify the right side:
-
Combine the quadratic terms: \[ 4x^2 - 14x^2 + 14x^2 = 4x^2 \]
-
Combine the linear terms: \[ 12x - 12x = 0 \]
-
Combine the constant terms: \[ 5 - 8 - 8 = -11 \]
So, the simplified equation becomes: \[ 21x^2 - 5x + 4 = 4x^2 - 11 + S \]
Now, isolate \( S \) by moving everything else to the left side: \[ S = (21x^2 - 5x + 4) - (4x^2 - 11) \]
Distributing the negative sign: \[ S = 21x^2 - 5x + 4 - 4x^2 + 11 \]
Combine like terms: \[ S = (21x^2 - 4x^2) + (-5x) + (4 + 11) = 17x^2 - 5x + 15 \]
Thus, the length of the missing side is: \[ \boxed{17x^2 - 5x + 15} \]