Determine the correct order of operations to solve the equation: (7) $5+\left(3 \cdot 4^{2}+2\right)$

Step 1 ：First, we need to solve the operation inside the parentheses, which includes an exponent and a multiplication operation. According to the order of operations in mathematics (PEMDAS), we should first calculate the exponent, then the multiplication, and finally the addition.

Step 2 ：Calculate the exponent: \(4^{2} = 16\)

Step 3 ：Then, multiply 3 by the result of the exponent: \(3 \cdot 16 = 48\)

Step 4 ：Add 2 to the result of the multiplication: \(48 + 2 = 50\)

Step 5 ：Finally, add 5 to the result of the operation inside the parentheses: \(5 + 50 = 55\)

Step 6 ：So, the final answer is \(\boxed{55}\)