Fractorise:
`9a^(2) -b ^(2)+4b-4`

Answer

To factorize the expression \(9a^2 - b^2 + 4b - 4\), we can follow these steps:

### Step 1: Rearrange the expression
We can rearrange the expression to group similar terms:
\[ 9a^2 - (b^2 - 4b + 4) \]

### Step 2: Recognize a perfect square
Notice that the expression inside the parentheses, \(b^2 - 4b + 4\), is a perfect square trinomial. It can be factored as:
\[ b^2 - 4b + 4 = (b - 2)^2 \]

### Step 3: Substitute back into the expression
Now, we substitute this back into our expression:
\[ 9a^2 - (b - 2)^2 \]

### Step 4: Apply the difference of squares formula
The expression \(9a^2 - (b - 2)^2\) is a difference of squares, which can be factored using the formula \(A^2 - B^2 = (A + B)(A - B)\). Here, \(A = 3a\) and \(B = (b - 2)\):
\[ 9a^2 - (b - 2)^2 = (3a + (b - 2))(3a - (b - 2)) \]

### Step 5: Simplify the factors
Simplifying the factors gives us:
\[ (3a + b - 2)(3a - b + 2) \]

### Final Factored Form
Thus, the final factored form of the expression \(9a^2 - b^2 + 4b - 4\) is:
\[ (3a + b - 2)(3a - b + 2) \]

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