https://file.aiquickdraw.com/mathbot/file/1725571244870mem6i9t0.jpg

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!

Simplify the expression: 4a^2b(a + b)^2 + 6ab^2(a + b)

Learn how to simplify the polynomial expression 4a^2b(a + b)^2 + 6ab^2(a + b) with a step-by-step solution. This problem involves key algebraic concepts like factorization and the distributive property, suitable for students in Grades 10-12.

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