Phân tích các đa thức sau thành nhân tử

a) x² – y² + 2x + 1;

b) (x² + 9)² – 36x²;

a) x² – y² + 2x + 1

= (x² + 2x + 1) – y²

= (x + 1)² – y²

= (x + 1 + y)(x + 1 – y).

b) (x² + 9)² – 36x²

= (x² + 9)² – (6x)²

= (x² + 9 – 6x)(x² + 9 + 6x)

= (x – 3)²(x + 3)².

c)$8{\text{x}}^3+\frac{1}{27}={\left(2\text{x}\right)}^3+{\left(\frac{1}{3}\right)}^3=\left(2\text{x}+\frac{1}{3}\right)\left[{\left(2\text{x}\right)}^2-2\text{x}.\frac{1}{3}+{\left(\frac{1}{3}\right)}^2\right]$

$=\left(2\text{x}+\frac{1}{3}\right)\left(4{\text{x}}^2-\frac{2}{3}x+\frac{1}{9}\right)$.

d) x³ – 8y³

= x³ – (2y)³

= (x – 2y)(x² + 2xy + 4y²).