Simplify (a+b)·4·(b+2): Applying the Distributive Property

Let's simplify the expression $(a+b)\cdot4\cdot(b+2)$ using the distributive property.

Step 1: Distribute the $4$ across $(b + 2)$.
$4(b + 2) = 4b + 4 \times 2 = 4b + 8$

Step 2: Now distribute this result $(4b + 8)$ across $(a + b)$.
$(a+b)(4b+8) = a(4b+8) + b(4b+8)$

Step 3: Apply the distributive property again for both terms.
- For $a(4b+8)$, we get:
$a \times 4b + a \times 8 = 4ab + 8a$
- For $b(4b+8)$, we get:
$b \times 4b + b \times 8 = 4b^2 + 8b$

Step 4: Combine all parts.
The expanded expression is:
$4ab + 8a + 4b^2 + 8b$

Therefore, the simplified expression is $\boxed{4ab + 8a + 4b^2 + 8b}$, and the correct choice is:

Yes, $4ab + 8a + 4b^2 + 8b$.