Algebra Factoring help?

Question

1) x^4 - 4x^2  +3

2)(x^2-1)^2 -1

Answer 1

1) The polynomial x⁴ - 4x² + 3

= x⁴ - x² - 3x² + 3

= x²(x² - 1) - 3(x² - 1)

= (x² - 1) (x² - 3)

= (x² - 1²) [x² - (√3)²]

Apply the formula (a² - b²) = (a - b) (a + b)

= ( x -1)(x + 1)(x + √3)(x - √3)

Factoring of x⁴ - 4x² + 3 is ( x -1)(x + 1)(x + √3)(x - √3).

Answer 2

2) The polynomial (x² - 1)² - 1

Apply the formula (a - b)² = a² + b² - 2ab

a = x², b = 1

(x² - 1)² - 1 = (x²)² + (1)² - 2(x²)(1) - 1

= x⁴ + 1 - 2x² - 1

= x⁴ -  2x²

= x²( x² -  2)

= x² [x² -  (√2)]²

= x² (x - √2)(x + √2)

Factoring of (x² - 1)² - 1 is (x)(x)(x - √2)(x + √2).