factorize the given expression `a^2+ab(b+1)+b^3`

Answer

To factorize the expression \( a^2 + ab(b + 1) + b^3 \), we can follow these steps:

### Step 1: Rewrite the Expression
Start by rewriting the expression clearly:
\[ a^2 + ab(b + 1) + b^3 \]

### Step 2: Expand the Expression
Now, expand the term \( ab(b + 1) \):
\[ ab(b + 1) = ab^2 + ab \]
So, the expression becomes:
\[ a^2 + ab^2 + ab + b^3 \]

### Step 3: Rearrange the Terms
Rearranging the terms, we have:
\[ a^2 + ab + ab^2 + b^3 \]

### Step 4: Group the Terms
Next, we can group the terms:
\[ (a^2 + ab) + (ab^2 + b^3) \]

### Step 5: Factor Out Common Terms
Now, we can factor out common terms from each group:
1. From the first group \( a^2 + ab \), we can factor out \( a \):
\[ a(a + b) \]
2. From the second group \( ab^2 + b^3 \), we can factor out \( b^2 \):
\[ b^2(a + b) \]

So now we have:
\[ a(a + b) + b^2(a + b) \]

### Step 6: Factor Out the Common Binomial
Now, notice that \( (a + b) \) is a common factor:
\[ (a + b)(a + b^2) \]

### Final Result
Thus, the factorized form of the expression \( a^2 + ab(b + 1) + b^3 \) is:
\[ (a + b)(a + b^2) \]