a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as \( n \). The result is \( 4 n \). (Simplify your answer.) b. Represent the original number as \( n \), and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 8 . Add 12 to the product.

To write a conjecture relating the result to the original number \( n \), we can observe that if we multiply \( n \) by 8, we get \( 8n \), and then adding 12 gives us \( 8n + 12 \). Now, if we represent the result as \( 4n \), the conjecture suggests that \( 8n + 12 \) ultimately simplifies to a relationship involving \( 4n \). Now, let's prove this conjecture using deductive reasoning. Starting with our original number \( n \), we perform the following steps: 1. Multiply by 8: \( 8n \) 2. Add 12: \( 8n + 12 = 4(2n + 3) \) This means that when we perform the operation of multiplying by 8 and then adding 12, the result can indeed be simplified to show a relationship with \( 4n \), specifically as \( 4(2n + 3) \). Hence, although our conjecture indicated a direct equivalence of \( 8n + 12 \) to \( 4n \), we demonstrated that both expressions can share \( n \) in their operations from the starting point.