The problem presents a right triangle with side lengths labeled in terms of $x$. Here’s how to approach the solution:

1. Identify Triangle Properties: In a right triangle, the Pythagorean Theorem applies: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

2. Assign Variables to Triangle Sides:

   • One leg of the triangle is labeled $28$.
   • The other leg is $2x + 8$.
   • The hypotenuse is $5x - 4$.

3. Set Up the Pythagorean Equation: Substitute these expressions into the Pythagorean theorem: $28^2 + (2x + 8)^2 = (5x - 4)^2$

4. Calculate and Expand:

   • First, calculate $28^2$: $28^2 = 784$
   • Then expand $(2x + 8)^2$: $(2x + 8)^2 = 4x^2 + 32x + 64$
   • Next, expand $(5x - 4)^2$: $(5x - 4)^2 = 25x^2 - 40x + 16$

5. Form the Equation: Substitute these into the Pythagorean formula: $784 + 4x^2 + 32x + 64 = 25x^2 - 40x + 16$

6. Combine Like Terms: Rearrange the terms to bring everything to one side of the equation: $0 = 21x^2 - 72x - 832$

7. Solve the Quadratic Equation: Simplify the equation by dividing each term by 7: $0 = 3x^2 - 72x - 118$

From here, we can apply the quadratic formula: $x = \frac{-(-72) \pm \sqrt{(-72)^2 - 4 \cdot 3 \cdot (-118)}}{2 \cdot 3}$ This calculation will yield the solution for $x$. Let me know if you would like further help in solving this quadratic equation or need more explanation on any step.

Would you like the detailed solution for the quadratic formula? Here are related questions for further exploration:

1. How do we determine which side is the hypotenuse in any right triangle?
2. Can you explain each term in the Pythagorean Theorem?
3. What other types of triangles can we analyze similarly?
4. How can we solve quadratic equations using methods other than the quadratic formula?
5. What applications does the Pythagorean theorem have in real-world problems?

Tip: Always simplify equations where possible before solving; it can make calculations easier and reduce errors.