The given expression is:

$4a^2b(a + b)^2 + 6ab^2(a + b)$

Let's simplify this expression step-by-step:

Step 1: Factor out the common terms

Both terms have common factors of $ab(a + b)$, so let's factor that out:

$4a^2b(a + b)^2 + 6ab^2(a + b) = ab(a + b) \left( 4a(a + b) + 6b \right)$

Step 2: Expand and simplify inside the parentheses

Expand the expression inside the parentheses:

$4a(a + b) + 6b = 4a^2 + 4ab + 6b$

Step 3: Combine the terms:

Now, combine like terms:

$ab(a + b) \left( 4a + 6b \right)$

So, the simplified expression is:

$ab(a + b) \left( 4a + 6b \right)$

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

Related Questions:

1. How would this expression change if the exponents were different?
2. What if there was a negative sign instead of addition?
3. Can you apply similar techniques to other expressions?
4. How can you visualize this factorization?
5. What happens if you substitute values for $a$ and $b$?

Tip:

When simplifying, always look for the highest common factor to make calculations easier!