R.H.S. = (–2) (3 – x) = – 6 + 2x
∴ x² – 2x = – 6 + 2x
⇒ x² – 2x + 6 – 2x = 0
⇒ x² – 4x + 6 = 0
∵ It is of the form ax² + bx + c = 0 where a = 1,b = –4, c = 6.
Hence, the given equation is a quadratic equation.

L.H.S. = (x – 1) (2x + 1)
= 2x² + x – 2x – 1
= 2x² – x – 1
R.H.S. = x(x + 5) = x² + 5x
∴ 2x² – x – 1 = x² + 5x
⇒ 2x² – x² – x – 5x – 1 = 0
⇒ x² – 6x – 1 = 0
∵ It is of the form ax² + bx + c = 0 where a = 1, b = –b, c = –1.
Hence, the given equation is a quadratic equation.

L.H.S. = (x + 1)² = x² + 2x + 1
R.H.S. = 2(x – 3) = 2x –6
∴ 6x² + 2x + 1 = 2x - 6
⇒ x² + 2x + 1–2x + 6 = 0
⇒ x²+ 7 = 0
∵ It is of the form ax² + bx + c = 0, where a = 1,b = 0, c = 7.
Hence, the given equation is a quadratic equation.

L.H.S. = (x–2) (x + 1)
= x² – x – 2
R.H.S. = (x – 1) (x + 3)
= x² + 2x – 3
∴ x² – x – 2 = x² + 2x – 3
⇒ – x –2 – 2x + 3 = 0
⇒ – 3x + 1 = 0
∵ It is not of the form ax² + bx + c = 0.
Hence, the given equation is not a quadratic equation. 

L.H.S. = (2x – 1) (x – 3)
= 2x² – 6x– + 3
= 2x² – 7x + 3
R.H.S. = (x + 5) (a – 1)
= x² + 5x – x – 5
= x² + 4x – 5
∴2x² – 7x + 3 = x² + 4x – 5
⇒ 2x² –x²-7x–4x+ 3 + 5 = 0
⇒ x² – 11x + 8 = 0
∵ It is of the form ax² + bx + c = 0 where a = 1, b = –11, c = 8.
Hence, the given equation is a quadratic equation.