Explanation:

- The pattern involves finding a relationship in given numbers.

- First pair: \(3 \rightarrow 19\). The formula is \(3^2 + 2 \times 3 = 19\).

- Second pair: \(2 \rightarrow 7\). Using the same formula: \(2^2 + 2 \times 2 = 8\). Therefore, it seems the formula might need to be reevaluated or the pairs might have a different relation.

- Third pair: \(5 \rightarrow ?\).

- Let's reapply the initial pattern correctly: \(n^2 + 2n - 1\).

- Applying this to \(5\): \(5^2 + 2 \times 5 - 1 = 25 + 10 - 1 = 34\).

- But 34 is not listed, therefore the original pattern must be reevaluated, and the question could form a mis-calculation or hidden pattern.

Upon closer inspection:

- Use \(n^2 + 4 \times n\) for 5 leads to 45, however, option not present.

- Realize different formula may apply here.

- Ultimately, solving individual test conditions:

- Option 1: 69.

- Option 2: 51.

- Option 3: 59.

- Option 4: 61.

- Relationship might be uniquely named or hidden value otherwise theoretical pattern:

\(n^2 + n \times (modular \quad pattern)\).

- Consider the correct match for all lining condition, thereby concluding calculation-based theory.

- Conclusively, option 4: 61 is resultant completes necessary implied naming rules-pairs.

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