The problem in the image reads:

"The sum of the squares of two consecutive even natural numbers is 100. Find the numbers."

Let's solve this step-by-step:

1. Let the two consecutive even natural numbers be $x$ and $x + 2$.
2. The square of $x$ is $x^2$ and the square of $x + 2$ is $(x + 2)^2$.
3. According to the problem, the sum of these squares is 100: $x^2 + (x + 2)^2 = 100$
4. Expanding $(x + 2)^2$: $x^2 + (x^2 + 4x + 4) = 100$
5. Combining like terms: $2x^2 + 4x + 4 = 100$
6. Subtracting 100 from both sides: $2x^2 + 4x - 96 = 0$
7. Dividing the entire equation by 2: $x^2 + 2x - 48 = 0$
8. Now, we can factor this quadratic equation: $(x + 8)(x - 6) = 0$
9. Setting each factor to zero gives: $x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad (\text{not a natural number})$ $x - 6 = 0 \quad \Rightarrow \quad x = 6$
10. So, the two consecutive even natural numbers are $6$ and $8$.

Answer: The two consecutive even natural numbers are 6 and 8.

Would you like further details or explanations on this solution?

Here are five related questions you might find helpful:

1. What are other methods to solve quadratic equations like factoring, completing the square, or using the quadratic formula?
2. How can we identify whether numbers are consecutive even numbers in a sequence?
3. How do we solve similar problems with consecutive odd numbers instead of even?
4. What are some practical applications of solving equations involving sums of squares?
5. How do we check if our solution satisfies the original problem conditions?

Tip: When solving problems involving consecutive numbers, assigning variables strategically (like $x$ and $x + 2$ for even numbers) helps simplify equations and ensure accuracy.