Simplify:

(i) 2x²(x³ – x) – 3x(x⁴ + 2x) – 2(x⁴ – 3x²)

= 2x⁵ – 2x³ – 3x⁵ – 6x² – 2x⁴ + 6x²

= -x⁵ – 2x⁴ – 2x³

(ii) x³y(x² – 2x) + 2xy(x³ – x⁴)

= x⁵y – 2x⁴y + 2x⁴y – 2x⁵y

= -x⁵y

(iii) 3a² + (a + 2) – 3a(2a + 1)
= 3a² + a + 2 – 6a² – 34

= -3a² – 2a + 2

(iv) x(x + 4) + 3x(2x² – 1) + 4x² + 4

= x² + 4x + 6x³ – 3x + 4x² + 4

= 6x³ + 5x² + x + 4

(v) a(b – c) – b(c – a) – c(a – b)

= ab – ac – bc + ab – ca + bc

= 2ab – 2ac

(vi) a(b – c) + b(c – a) + c(a – b)

= ab – ac + bc – ab + ac – bc

= 0

(vii) 4ab(a – b) – 6a²(b – b²) – 3b²(2a² – a) + 2ab(b – a)

= 4a²b – 4ab² – 6a²b + 6a²b² – 6a²b² + 3ab² + 2ab² – 2a²b

= 3ab²

(viii) x²(x² + 1) – x³(x + 1) – x(x³ – x)

= x⁴ + x² – x⁴ – x³ – x⁴ + x²

= 2x² – 2x³

(ix) 2a² + 3a (1 – 2a³) + a(a + 1)

= 2a² + 3a – 6a⁴ + a² + a

= -6a⁴ + 3a² + 4a

(x) a²(2a – 1) + 3a + a³ – 8

= 2a³ – a² + 3a + a³ – 8

= 3a³ – a² + 3a – 8



(xii) a²b(a – b²) + ab²(4ab – 2a²) – a³b(1 – 2b)

= a³b – a²b³ + 4a²b³ – 2a³b² – a³b + 2a³b²

= -a²b³ + 4a²b³

= 3a²b³

(xiii) a²b(a³ – a + 1) – ab(a⁴ – 2a² + 2a) – b(a³ – a² – 1)

= a⁵b – a³b + a²b – a⁵b + 2a³b – 2a²b – ba³ + a²b + b

= b