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

Answer

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

### Step 1: Expand the expression
First, we need to expand the term \( ab(1 - 2a) \):
\[ ab(1 - 2a) = ab - 2a^2b \]
Now, substitute this back into the expression:
\[ a^3 + ab - 2a^2b - 2b^2 \]

### Step 2: Rearrange the expression
Now, let's rearrange the terms:
\[ a^3 - 2a^2b + ab - 2b^2 \]

### Step 3: Group the terms
Next, we can group the first two terms and the last two terms:
\[ (a^3 - 2a^2b) + (ab - 2b^2) \]

### Step 4: Factor out common terms
Now, we can factor out common terms from each group:
1. From \( a^3 - 2a^2b \), we can factor out \( a^2 \):
\[ a^2(a - 2b) \]
2. From \( ab - 2b^2 \), we can factor out \( b \):
\[ b(a - 2b) \]

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

### Step 5: Factor out the common binomial factor
Now, we can see that \( (a - 2b) \) is a common factor:
\[ (a - 2b)(a^2 + b) \]

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

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