To find the perimeter of the triangle given the lengths of its sides represented by polynomials, we need to sum the expressions for the lengths of the three sides.
The side lengths are given as follows:
- \( \frac{21}{2}x \)
- \( \frac{5}{4}x - 3 \)
- \( \frac{47}{4}x \)
Now, we will calculate the perimeter \( P \) of the triangle by adding these three expressions:
\[ P = \left( \frac{21}{2}x \right) + \left( \frac{5}{4}x - 3 \right) + \left( \frac{47}{4}x \right) \]
Let's start by combining the \( x \) terms. First, we need to convert \( \frac{21}{2}x \) into a fraction with a denominator of 4:
\[ \frac{21}{2} = \frac{21 \times 2}{2 \times 2} = \frac{42}{4} \]
Now rewriting the expressions with a common denominator:
\[ P = \left( \frac{42}{4}x \right) + \left( \frac{5}{4}x - 3 \right) + \left( \frac{47}{4}x \right) \]
Combining the \( x \) terms:
\[ P = \left( \frac{42}{4}x + \frac{5}{4}x + \frac{47}{4}x \right) - 3 \] \[ = \left( \frac{42 + 5 + 47}{4}x \right) - 3 \] \[ = \left( \frac{94}{4}x \right) - 3 \] \[ = \frac{47}{2}x - 3 \]
Thus, the expression that represents the perimeter of the triangle in units is:
\[ \frac{47}{2}x - 3 \]
Since this specific expression wasn’t listed among the provided options, we can conclude that the entire calculation and method are correct for finding the perimeter from the given sides, but the final result does not exactly fit into any of the expressions listed. Therefore, if we were to strictly follow the options given, none correctly give the complete form of the perimeter calculated.