factorize : `4a^2-4b^2+4a+1`

Explanation

To factorize the expression \(4a^2 - 4b^2 + 4a + 1\), we can follow these steps:

### Step 1: Rearrange the expression
We can rearrange the expression to group similar terms:
\[ 4a^2 + 4a - 4b^2 + 1 \]

### Step 2: Factor out common terms
Notice that \(4a^2 + 4a\) can be factored:
\[ 4(a^2 + a) - 4b^2 + 1 \]

### Step 3: Complete the square for \(a\)
To complete the square for \(a^2 + a\), we can rewrite it as:
\[ a^2 + a = \left(a + \frac{1}{2}\right)^2 - \frac{1}{4} \]
Thus, we can express \(4(a^2 + a)\) as:
\[ 4\left(\left(a + \frac{1}{2}\right)^2 - \frac{1}{4}\right) = 4\left(a + \frac{1}{2}\right)^2 - 1 \]

### Step 4: Substitute back into the expression
Now substituting back into the expression gives:
\[ 4\left(a + \frac{1}{2}\right)^2 - 1 - 4b^2 \]
This can be rearranged to:
\[ 4\left(a + \frac{1}{2}\right)^2 - (2b)^2 \]

### Step 5: Apply the difference of squares formula
Now we can use the difference of squares formula \(A^2 - B^2 = (A + B)(A - B)\):
Let \(A = 2\left(a + \frac{1}{2}\right)\) and \(B = 2b\). Thus, we have:
\[ (2\left(a + \frac{1}{2}\right) + 2b)(2\left(a + \frac{1}{2}\right) - 2b) \]

### Step 6: Simplify the factors
This simplifies to:
\[ (2a + 1 + 2b)(2a + 1 - 2b) \]

### Final Factorized Form
Thus, the factorized form of the expression \(4a^2 - 4b^2 + 4a + 1\) is:
\[ (2a + 1 + 2b)(2a + 1 - 2b) \]

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