1. The problem is to express "decreased by 8 times Y plus three" as a mathematical expression. 2. "Decreased by" means subtraction. 3. "8 times Y" means multiplication: $8Y$. 4. "Plus three" means adding 3. 5. Putting it all together, if we start with a number $x$, then "decreased by 8 times Y plus three" means: $$x - (8Y + 3)$$ 6. This expression shows that from $x$, we subtract the quantity $8Y + 3$. 7. The parentheses are important to indicate that both $8Y$ and $3$ are subtracted together. Final answer: $$x - (8Y + 3)$$