Factorize following expressions
(i) `a^2-4a+3+2b-b^2" (ii)" x^(4)+324" (iii)"x^4-y^2+2x^2+1`
(iv) `4a^4-5a^2+1" (v)"x^6-7x^2-6`

Answer

Let's solve the expressions step by step.

### (i) Factorize \( a^2 - 4a + 3 + 2b - b^2 \)

1. **Rearranging the expression**:
\[ a^2 - 4a + (2b - b^2) + 3 \]
This helps us see the quadratic terms more clearly.

2. **Completing the square for \( a^2 - 4a + 3 \)**:
\[ a^2 - 4a + 4 - 1 = (a - 2)^2 - 1 \]

3. **Rearranging \( 2b - b^2 \)**:
\[ 2b - b^2 = -(b^2 - 2b) = -(b^2 - 2b + 1 - 1) = -(b - 1)^2 + 1 \]

4. **Combining the completed squares**:
\[ (a - 2)^2 - 1 - (b - 1)^2 + 1 = (a - 2)^2 - (b - 1)^2 \]

5. **Using the difference of squares formula**:
\[ (x^2 - y^2 = (x - y)(x + y)) \]
Here, \( x = (a - 2) \) and \( y = (b - 1) \):
\[ (a - 2 - (b - 1))(a - 2 + (b - 1)) = (a - b + 1)(a + b - 3) \]

**Final Factorization**:
\[ (a - b + 1)(a + b - 3) \]