To expand and simplify the expression $2(4d + 5)(3d + 1)$, let's go through the steps:

1. Expand $(4d + 5)(3d + 1)$: $(4d + 5)(3d + 1) = 4d \cdot 3d + 4d \cdot 1 + 5 \cdot 3d + 5 \cdot 1$ Breaking this down:

   • $4d \cdot 3d = 12d^2$
   • $4d \cdot 1 = 4d$
   • $5 \cdot 3d = 15d$
   • $5 \cdot 1 = 5$

   Now combine these results: $(4d + 5)(3d + 1) = 12d^2 + 4d + 15d + 5$ Combine like terms: $= 12d^2 + 19d + 5$

2. Multiply by 2: Now, we distribute the 2 across each term: $2(12d^2 + 19d + 5) = 24d^2 + 38d + 10$

So, the expanded and simplified form of the expression is: $\boxed{24d^2 + 38d + 10}$

Would you like further details on any steps or have other questions?

Here are five related questions:

1. How would you approach a similar problem with three binomials?
2. What would change if one of the terms was negative, such as $(4d - 5)(3d + 1)$?
3. Can you factor the result back into its original terms?
4. How can you verify the expanded expression is correct?
5. How would this change if there was a variable multiplier instead of the number 2?

Tip: Remember to combine like terms after each distribution step to keep your work organized and simplified.