Phân tích đa thức thành nhân tử

a) x³y + xy - y - 1
= y(x³ + x) - 1
= y(x(x² + 1)) - 1
= yx(x² + 1) - 1

b) x²(x - 2) + 4(2 - x)
= x²(x - 2) - 4(x - 2)
= (x - 2)(x² - 4)
= (x - 2)(x + 2)(x - 2)

c) x³ - x² - 20x
= x(x² - x - 20)
= x(x - 5)(x + 4)

d) (x² + 1)² - (x + 1)²
= [(x² + 1) - (x + 1)][(x² + 1) + (x + 1)]
= (x² - x)(x² + x + 2)

e) 6x² - 7x + 2
= (2x - 1)(3x - 2)

f) x⁴ + 8x² + 12
= (x² + 4)(x² + 3)

g) (x³ + x + 1)(x³ + x) - 2
= (x³ + x)² + (x³ + x) - 2
= t² + t - 2 (đặt t = x³ + x)
= (t + 2)(t - 1)

h) (x + 1)(x + 2)(x + 3)(x + 4) + 1
= ((x + 1)(x + 4))((x + 2)(x + 3)) + 1
= (x² + 5x + 4)(x² + 5x + 6) + 1

i) - (x² + 2)² + 4x(x² + 2) - 3x²
= - (x² + 2)(x² + 2) + 4x(x² + 2) - 3x²
= - A² + 4Ax - 3x² with A = x² + 2
= A(4x - A - 3)

j) 8x⁴ + 4y⁴
= 4(2x⁴ + y⁴)
= 4[(x²)² + (y²)²]
= 4(x² + y²)((x²)² - (y²)²)
= 4(x² + y²)(x² - y²)(x² + y²)