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.

If an original number \( n \) is multiplied by 8 and then 12 is added to the product, the result can be expressed as \( 8n + 12 \). This can be simplified to \( 4(2n + 3) \), leading us to the conjecture that the result is four times \( (2n + 3) \), closely associating it with the original number selected. To prove this, let's start with the original number \( n \). When multiplied by 8, we get \( 8n \). Next, adding 12 gives us \( 8n + 12 \). If we factor out the 4 from \( 8n + 12 \), we rewrite it as \( 4(2n + 3) \). Thus, our conjecture holds true because \( 8n + 12 \) simplifies cleanly to four times another expression related to the original number.