Algebra combining like terms puzzle in a flower design.

Problem Analysis:

The image shows a geometric pattern with various algebraic expressions and numerical values. The task appears to involve solving for the variable \( x \) using the relationships between these expressions. The pattern is symmetrical, and the expressions are repeated in a specific manner.

Key Observations:

1. Repeating Expressions: The expressions \( 3x^2 \), \( x^2 \), \( 14x \), \( -10x \), \( 4x \), \( 2x \), \( -18 \), and \( -20 \) appear repeatedly.
2. Symmetry: The pattern is highly symmetric, suggesting that the relationships between the expressions might be consistent across different sections.
3. Numerical Values: The numbers \( 6 \) and constants like \( -18 \) and \( -20 \) are also present, which might help in forming equations.

Step-by-Step Solution:

#### Step 1: Identify Relationships
The problem likely involves setting up equations based on the relationships between the expressions. Let's focus on one section of the pattern and generalize.

#### Step 2: Choose a Section
Consider one of the central hexagonal regions. For example, the topmost hexagon contains the following expressions:
- Top: \( 3x^2 \)
- Left: \( -10x \)
- Bottom-left: \( -18 \)
- Bottom-right: \( -20 \)
- Right: \( 14x \)

#### Step 3: Form Equations
We need to find a relationship between these expressions. A common approach in such problems is to assume that the sum of the expressions around a central point or region equals zero (a common constraint in similar puzzles).

Let's assume the sum of the expressions around the topmost hexagon is zero:
\[
3x^2 + (-10x) + (-18) + (-20) + 14x = 0
\]

#### Step 4: Simplify the Equation
Simplify the equation step by step:
\[
3x^2 - 10x - 18 - 20 + 14x = 0
\]
Combine like terms:
\[
3x^2 + (14x - 10x) - 38 = 0
\]
\[
3x^2 + 4x - 38 = 0
\]

#### Step 5: Solve the Quadratic Equation
We now have a quadratic equation:
\[
3x^2 + 4x - 38 = 0
\]
We can solve this using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = 4 \), and \( c = -38 \).

Calculate the discriminant:
\[
b^2 - 4ac = 4^2 - 4 \cdot 3 \cdot (-38) = 16 + 456 = 472
\]

Now, apply the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{472}}{2 \cdot 3} = \frac{-4 \pm \sqrt{472}}{6}
\]

Simplify \( \sqrt{472} \):
\[
\sqrt{472} = \sqrt{4 \cdot 118} = 2\sqrt{118}
\]

Thus:
\[
x = \frac{-4 \pm 2\sqrt{118}}{6} = \frac{-2 \pm \sqrt{118}}{3}
\]

#### Step 6: Determine the Valid Solution
The solutions are:
\[
x = \frac{-2 + \sqrt{118}}{3} \quad \text{and} \quad x = \frac{-2 - \sqrt{118}}{3}
\]

Since the problem does not specify any constraints on \( x \), both solutions are mathematically valid. However, typically, problems of this nature expect a single solution, so we might need additional context to choose between them. Without further context, we present both solutions.

Final Answer:

\[
\boxed{\frac{-2 + \sqrt{118}}{3}}
\]