Let's go through the process of expanding and simplifying the expression $2(4d + 5)(3d + 1)$.

Step 1: Apply the distributive property (also known as FOIL for binomials)

We first expand the product $(4d + 5)(3d + 1)$. Using distributive property (multiply each term in the first binomial by each term in the second binomial):

$(4d + 5)(3d + 1) = 4d(3d) + 4d(1) + 5(3d) + 5(1)$ $= 12d^2 + 4d + 15d + 5$

Step 2: Combine like terms

Now, combine the like terms:

$12d^2 + (4d + 15d) + 5 = 12d^2 + 19d + 5$

Step 3: Multiply by 2

Finally, we distribute the 2 across the terms:

$2(12d^2 + 19d + 5) = 24d^2 + 38d + 10$

Final Answer:

The expanded and simplified form of $2(4d + 5)(3d + 1)$ is:

$24d^2 + 38d + 10$

Would you like further details or have any questions on this process?

Here are five questions related to this topic:

1. How does the distributive property work in algebra?
2. Can you expand and simplify expressions with more than two binomials?
3. What happens if you have exponents in the binomials being multiplied?
4. What is the difference between expanding and factoring expressions in algebra?
5. How do you recognize and combine like terms?

Tip: Always watch for common factors in your expressions before expanding. Simplifying early can sometimes make the process quicker.