Find the lowest common multiple of the following expressions:

x² – 3x + 2 and x⁴ + x³ – 6x²

Let u(x) = x² – 3x + 2

On factorizing the above equation,

Sum = -3

Product = 2

Therefore the two numbers satisfying the above conditions are -2 and -1.

u(x) = x² – x – 2x – 2

u(x) = x(x – 1) - 2(x – 1)

u(x) = (x – 2) (x – 1)

Let v(x) = x⁴ + x³ – 6x²

= x²(x² + x – 6)

Solving the quadratic part,

⟹ x² + x – 6

⟹ x² + 3x – 2x – 6

⟹ x(x + 3) – 2(x + 3)

⟹ (x – 2) (x + 3)

Therefore v (x) = x² (x – 2) (x + 3)

By comparing u(x) and v(x),

LCM = x² (x – 2) (x + 3) (x – 1)