(i) (x + 1)^{2} = 2(x – 3)

x^{2} + 2x + 1 = 2x – 6

x^{2} + 2x – 2x +1 + 6 = 0

x^{2} + 0x + 7 = 0

This is in the form of ax^{2} + bx + c = 0.

∴ This is a quadratic equation.

(ii) x^{2} – 2x = (-2) (3 – x)

x^{2} – 2x = -6 + 2

x^{2} – 2x – 2x + 6 = 0

x^{2} – 4x + 6 = 0

This is in the form of ax^{2} + bx + c = 0.

∴ This is a quadratic equation.

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

x^{2} + x – 2x – 2 = x^{2} + 3x – x – 3

x^{2} – x^{2} = x^{2} + 2x – 3

x^{2} – x^{2} – x – 2x – 2 + 3 = 0

-x – 2x – 2 + 3 = 0

-3x + 1 = 0

3x – 1 = 0

This is not in the form of ax^{2} + bx + c = 0

∴ Given equation is not quadratic equation.

(iv) (x – 3) (2x + 1) = x (x + 5)

2x^{2} + x – 6x – 3 = x^{2} + 5x

2x^{2} – 5x – 3 = x^{2} + 5x

2x^{2} – x^{2} – 5x – 5x – 3 = 0

x^{2} – 10x – 3 = 0

This is in the form of ax^{2} + bx + c = 0.

∴ This is a quadratic equation.

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

2x^{2} – 6x – x + 3 = x^{2} – x + 5x – 5

2x^{2} – 7x + 3 = x^{2} + 4x – 5

2x^{2} – x^{2} – 7x – 4x + 3 + 5 = 0

x^{2} – 11x + 8 = 0

This is in the form of ax^{2} + bx + c = 0.

∴ This is a quadratic equation.

(vi) x^{2} + 3x + 1 = {x – 2}^{2}

x^{2} + 3x + 1 = x^{2 }– 4x + 4

x^{2} + 3x + 1 – x^{2} + 4x – 4 = 0

7x – 3 = 0

This is not in the form of ax^{2} + bx + c = 0

∴ Given equation is not quadratic equation.

(vii) (x + 2)^{3} = 2x (x^{2} – 1)

x^{3} + 8 + 3(x)(2)(x+2) = 2x^{3 }– 2x

x^{3} + 8 = 6x(x + 2) = 2x^{3} – 2x

x^{3} + 8 + 6x^{2} + 12x = 2x^{3} – 2x

x^{3} – 2x^{3} + 6x^{2} + 12x + 2x + 8 = 0

-x^{3 }+ 6x^{2} + 14x + 8 = 0

x^{3} – 6x^{2} – 14x – 8 = 0

This is not in the form of ax^{2} + bx + c = 0

∴ Given equation is not quadratic equation.

(viii) x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}

x^{3} – 4x^{2} – x + 1 = x^{3} – 8 – 3(x)(2)(x – 2)

x^{3} – 4x^{2} – x + 1 = x^{2} – 8 – 6x (x – 2)

x^{3} – 4x^{2} – x + 1 = x^{3 }– 8 – 6x^{2} + 12x

x^{3} – x^{3} – 4x^{2} + 6x^{2} – x – 12x + 1 + 8 = 0

+2x^{2 }– 13x + 9 = 0

This is in the form of ax^{2} + bx + c = 0.

∴ This is a quadratic equation.

(x+1)^2 =2(x-3)…x^2 +1 + 2×x×1 = 2x – 6x^2 +2x-2x+1+6 =0x^2+0x+7=0so ,it is a quadratic equation as a=1,b=0 nd c=7