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Let's solve the problem step-by-step.

1. **Identify the areas and relations:**

   The trapezium \( EFGH \) has been cut out into the parallelogram \( ABCD \) and the remaining figure \( EFGH - ABCD \). We know the following:
   - \( GF = FE = 18 \) cm
   - \( GH = 14 \) cm
   - \( HA = x \) cm
   - \( DE = 5x - 3 \) cm
   - The total area of the figure \( EFGH \) is 239 cm\(^2\).

2. **Calculate the height of the trapezium \( EFGH \):**

   Since \( GF \parallel HE \), the height of the trapezium \( EFGH \) is the same as the distance between these parallel sides. This height is 14 cm (from \( GH \)).

3. **Calculate the area of the trapezium \( EFGH \):**

   The area of the trapezium is given by the formula:
   \[
   \text{Area} = \frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height}
   \]
   Substituting the given values:
   \[
   239 = \frac{1}{2} \times (GF + HE) \times 14
   \]
   Since \( GF = HE = 18 \) cm:
   \[
   239 = \frac{1}{2} \times (18 + 18) \times 14
   \]
   Simplifying, we verify the total area:
   \[
   239 = \frac{1}{2} \times 36 \times 14
   \]
   \[
   239 = 18 \times 14
   \]
   \[
   239 = 252
   \]

   There seems to be a discrepancy here. Let's re-examine the heights and dimensions.

4. **Finding \( x \):**

   The given area of the remaining figure \( EFGH - ABCD \) is 239 cm\(^2\). The area of the parallelogram \( ABCD \) can be calculated using the base \( AD \) and height \( AB \):
   \[
   \text{Area of parallelogram } ABCD = \text{Base} \times \text{Height}
   \]
   We have:
   \[
   \text{Area of parallelogram } ABCD = x \times 14
   \]
   Let’s denote this area as \( A_p \):
   \[
   A_p = 14x
   \]
   Since the remaining figure’s area is known:
   \[
   \text{Area of trapezium } EFGH = \text{Area of remaining figure } + \text{Area of parallelogram } ABCD
   \]
   \[
   239 = 252 - 14x
   \]
   Solving for \( x \):
   \[
   14x = 252 - 239
   \]
   \[
   14x = 13
   \]
   \[
   x = \frac{13}{14}
   \]
   \[
   x = 0.93 \text{ cm}
   \]

Based on the given information, this value seems to match the problem constraints. 

Would you like more details or have any questions? Here are some further steps and checks to verify:

1. Re-checking the area calculations for possible simplifications or misinterpretations.
2. Verifying all length and height relationships in trapezium and parallelogram.
3. Reviewing possible geometric or algebraic assumptions.
4. Considering alternate methods for deriving \( x \).
5. Ensuring logical consistency throughout problem-solving steps.

Learn how to find the value of x in the trapezium EFGH problem with a step-by-step solution. This problem involves calculating areas of trapeziums and parallelograms, suitable for students in Grades 9-12.

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