Diagram of a rhombus with diagonals and side lengths labeled.

We are given a geometric figure that appears to be a kite (a quadrilateral with two pairs of adjacent equal sides), labeled with points $ L, M, P, Q $, and some side lengths. Let's analyze the image and solve for $ x $.

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Step 1: Understand the Figure

From the diagram:

- $ LM = 15 $
- $ LP = 15 $
- $ MQ = 11 $
- $ PQ = 4x + 3 $
- Diagonals: $ MP $ and $ LQ $ intersect at point $ O $
- There is a right angle at $ O $, where diagonals intersect — this indicates that the diagonals are perpendicular.
- The diagonal $ LQ $ is vertical, and $ MP $ is horizontal.
- $ O $ is the intersection point of the diagonals.

This shape is a kite, because:
- Two pairs of adjacent sides are equal: $ LM = LP = 15 $, and $ MQ = PQ $? Wait — not necessarily yet.

But we have:
- $ LM = 15 $
- $ LP = 15 $
- $ MQ = 11 $
- $ PQ = 4x + 3 $

So unless $ 4x + 3 = 11 $, then $ MQ \neq PQ $. But in a kite, typically two pairs of adjacent equal sides are present.

Let’s look again:

- $ LM = LP = 15 $ → So triangle $ LMP $ is isosceles
- $ MQ = 11 $, $ PQ = 4x + 3 $

But if $ MQ $ and $ PQ $ are the other pair of adjacent sides from $ Q $, then for it to be a kite, either:
- $ MQ = PQ $, or
- $ MQ = something else $

Wait — actually, the vertices are $ L, M, P, Q $, so the sides are:
- $ LM $
- $ MP $
- $ PQ $
- $ QL $

But from the diagram, it looks like the kite has vertices in order: $ L, M, Q, P $, forming a symmetric shape.

But more importantly, notice that:

- $ LM = LP = 15 $ → So point $ L $ is equidistant to $ M $ and $ P $
- $ MQ = 11 $
- $ PQ = 4x + 3 $
- The diagonals $ MP $ and $ LQ $ intersect at right angles at point $ O $

In a kite, one diagonal is a line of symmetry. Here, since $ LM = LP $, and $ MQ $ and $ PQ $ may or may not be equal, but the symmetry suggests that $ LQ $ is the axis of symmetry.

Also, in a kite:
- One diagonal is perpendicular to the other.
- The diagonal connecting the equal adjacent sides (here, $ LM = LP $) is the axis of symmetry.

So $ LQ $ is the axis of symmetry, meaning:
- It bisects the other diagonal $ MP $ at right angles.
- And it reflects $ M $ onto $ P $, and $ Q $ onto itself.

So, by symmetry:
- $ MQ = PQ $

That’s key!

Because $ LQ $ is the axis of symmetry, and $ M $ and $ P $ are symmetric across $ LQ $, then:
$$
MQ = PQ
$$

We are told:
- $ MQ = 11 $
- $ PQ = 4x + 3 $

So set them equal:
$$
4x + 3 = 11
$$

Now solve for $ x $:

$$
4x = 11 - 3 = 8 \\
x = \frac{8}{4} = 2
$$

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✔ Final Answer:

$$
\boxed{x = 2}
$$

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Explanation Summary:

- The figure is a kite with $ LM = LP = 15 $, so $ L $ is the vertex of the two equal sides.
- The diagonal $ LQ $ is the axis of symmetry.
- Therefore, $ MQ = PQ $ due to symmetry.
- Given $ MQ = 11 $ and $ PQ = 4x + 3 $, we set up the equation:
$$
4x + 3 = 11
$$
- Solving gives $ x = 2 $.

✔ This makes sense geometrically and algebraically.