(i) (x + 1)² = 2(x – 3) 

x² + 2x + 1 = 2x – 6 

x² + 2x – 2x +1 + 6 = 0 

x² + 0x + 7 = 0 

This is in the form of ax² + bx + c = 0. 

∴ This is a quadratic equation. 

(ii) x² – 2x = (-2) (3 – x) 

x² – 2x = -6 + 2 

x² – 2x – 2x + 6 = 0 

x² – 4x + 6 = 0 

This is in the form of ax² + bx + c = 0. 

∴ This is a quadratic equation. 

(iii) (x – 2) (x + 1) = (x – 1) (x + 3) 

x² + x – 2x – 2 = x² + 3x – x – 3 

x² – x² = x² + 2x – 3 

x² – x² – x – 2x – 2 + 3 = 0 

-x – 2x – 2 + 3 = 0 

-3x + 1 = 0 

3x – 1 = 0 

This is not in the form of ax² + bx + c = 0 

∴ Given equation is not quadratic equation.

(iv) (x – 3) (2x + 1) = x (x + 5) 

2x² + x – 6x – 3 = x² + 5x 

2x² – 5x – 3 = x² + 5x 

2x² – x² – 5x – 5x – 3 = 0 

x² – 10x – 3 = 0 

This is in the form of ax² + bx + c = 0. 

∴ This is a quadratic equation. 

(v) (2x – 1) (x – 3) = (x + 5) (x – 1) 

2x² – 6x – x + 3 = x² – x + 5x – 5 

2x² – 7x + 3 = x² + 4x – 5 

2x² – x² – 7x – 4x + 3 + 5 = 0 

x² – 11x + 8 = 0 

This is in the form of ax² + bx + c = 0. 

∴ This is a quadratic equation. 

(vi) x² + 3x + 1 = {x – 2}² 

x² + 3x + 1 = x² – 4x + 4 

x² + 3x + 1 – x² + 4x – 4 = 0 

7x – 3 = 0 

This is not in the form of ax² + bx + c = 0 

∴ Given equation is not quadratic equation. 

(vii) (x + 2)³ = 2x (x² – 1) 

x³ + 8 + 3(x)(2)(x+2) = 2x³ – 2x 

x³ + 8 = 6x(x + 2) = 2x³ – 2x 

x³ + 8 + 6x² + 12x = 2x³ – 2x 

x³ – 2x³ + 6x² + 12x + 2x + 8 = 0 

-x³ + 6x² + 14x + 8 = 0 

x³ – 6x² – 14x – 8 = 0 

This is not in the form of ax² + bx + c = 0 

∴ Given equation is not quadratic equation. 

(viii) x³ – 4x² – x + 1 = (x – 2)³ 

x³ – 4x² – x + 1 = x³ – 8 – 3(x)(2)(x – 2) 

x³ – 4x² – x + 1 = x² – 8 – 6x (x – 2) 

x³ – 4x² – x + 1 = x³ – 8 – 6x² + 12x 

x³ – x³ – 4x² + 6x² – x – 12x + 1 + 8 = 0 

+2x² – 13x + 9 = 0 

This is in the form of ax² + bx + c = 0. 

∴ This is a quadratic equation.