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To find the correct expression for the perimeter of the quadrilateral, we need to sum up all its side lengths.

The side lengths given are:
- \( 5x - 3 \)
- \( 6x + 1 \) (twice, since there are two sides with this length)
- \( 11x + 10 \)

Now, let's calculate the perimeter by adding these expressions:

\[
\text{Perimeter} = (5x - 3) + (6x + 1) + (6x + 1) + (11x + 10)
\]

First, combine the like terms (all the \(x\) terms together and all the constant terms together):

\[
\text{Perimeter} = 5x + 6x + 6x + 11x - 3 + 1 + 1 + 10
\]

Simplify the expression:

\[
\text{Perimeter} = (5x + 6x + 6x + 11x) + (-3 + 1 + 1 + 10)
\]

\[
\text{Perimeter} = 28x + 9
\]

The expression that represents the perimeter is \(28x + 9\).

So, the correct answer is **C**.

Would you like any further details or have any other questions?

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Here are 5 questions to consider based on this explanation:

1. What are the steps to combine like terms in an algebraic expression?
2. How do you determine the perimeter of any polygon given its side lengths?
3. Why is it important to check each side length in a geometry problem?
4. How would the expression change if one of the side lengths was modified?
5. What are common errors to avoid when summing algebraic expressions?

**Tip:** Always double-check your addition of like terms, as a small mistake in combining them can lead to an incorrect solution.

The diagram shows the side lengths of a quadrilateral. Which expression represents the perimeter of the figure?

This problem demonstrates how to find the perimeter of a quadrilateral with algebraic side lengths by summing the expressions for each side. This is a useful exercise for students in Grades 8-10, covering key algebraic and geometric concepts.

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