To solve the given right triangle problem:

Triangle Setup:

• Hypotenuse = $2x + 2$
• One leg = $2x$
• Other leg = $2x - 2$

We will use the Pythagorean theorem: $a^2 + b^2 = c^2$ where $a = 2x - 2$, $b = 2x$, and $c = 2x + 2$.

Step 1: Apply the Pythagorean theorem

$(2x - 2)^2 + (2x)^2 = (2x + 2)^2$

Step 2: Expand each term

1. $(2x - 2)^2 = 4x^2 - 8x + 4$
2. $(2x)^2 = 4x^2$
3. $(2x + 2)^2 = 4x^2 + 8x + 4$

Substitute these into the equation: $(4x^2 - 8x + 4) + 4x^2 = 4x^2 + 8x + 4$

Step 3: Simplify the equation

Combine like terms: $4x^2 + 4x^2 - 8x + 4 = 4x^2 + 8x + 4$ $8x^2 - 8x + 4 = 4x^2 + 8x + 4$

Simplify further: $8x^2 - 4x^2 - 8x - 8x + 4 - 4 = 0$ $4x^2 - 16x = 0$

Step 4: Factorize the equation

$4x(x - 4) = 0$

Step 5: Solve for $x$

$x = 0 \quad \text{or} \quad x = 4$

Since $x = 0$ is not valid for side lengths, we have: $x = 4$

Step 6: Find the side lengths

Substitute $x = 4$ into the expressions for the sides:

1. Hypotenuse ($c$): $2x + 2 = 2(4) + 2 = 10$
2. One leg ($b$): $2x = 2(4) = 8$
3. Other leg ($a$): $2x - 2 = 2(4) - 2 = 6$

Final Answer:

The triangle has side lengths: $6, 8, \text{and } 10$

Would you like further clarification? Here are five related questions:

1. What happens if we try different values of $x$?
2. How does the Pythagorean theorem guarantee right triangle properties?
3. Can we use trigonometric ratios to verify the side lengths?
4. What other methods can determine the side lengths in such problems?
5. How does factoring help simplify quadratic equations?

Tip: Always check for extraneous solutions when solving quadratic equations!