The pattern in the given analogy is based on a mathematical relationship between pairs of numbers. Let's analyze the relationship between the first pair (84 : 196) and the third pair (96 : 224):

• 84 : 196: Notice that \(196 = 84 \times 2 + 28\). The second number is double the first number plus 28.
• 96 : 224: Similarly, \(224 = 96 \times 2 + 32\). The second number is double the first number plus 32.

Observe that the added constant increases by 4 (from 28 to 32) as we move from the first pair to the second pair. This suggests a pattern.

Now, let's apply this pattern to the second pair (51 : x) to find the missing number (x):

If the pattern continues, the added constant should be \(32 + 4 = 36\).

Therefore, \(x = 51 \times 2 + 36 = 102 + 36 = 138\).

However, this result is not among the options provided. Let's re-examine the pattern. It seems there is another logical pattern present.

Let's try another approach. Consider the difference between the numbers in each pair:

• 196 - 84 = 112
• 224 - 96 = 128

The difference increases by 16 (128 - 112 = 16). If we maintain this pattern, the difference for the middle pair should be 112 + 16 = 128

Therefore, x = 51 + 112 + 16 = 179. This is also not an option. Let's consider another pattern.

Let's examine the relationship in a different light. If we look at the prime factorization of the numbers, there's no immediately apparent pattern. Let's try a simpler approach. Let's consider if one number is a multiple of the other.

It turns out that \(196 = 84 \times 2 + 28\) and \(224 = 96 \times 2 + 32\). While not a clean ratio, the relationship between the pairs involves multiplication by 2 and adding an increasing constant. This is a subtle pattern.

There might be an error in the question or the provided options. None of the simple arithmetic relationships yield a consistent pattern that leads to one of the provided options. The explanation assumes a potential error in the question or answer choices.