1)
a)
Given
x ^{2} - 3 x + 2 = 0
Identify the coefficients a,b and c.
a = 1, b = - 3 , c = 2
Find the the discriminant
D = b^{ 2} - 4 a c = (-3)^{ 2} - 4 (1) (2) = 1.
Since the discriminant is positive, the above equation has two real solutions
x_{1} = (-b + √D) / (2 a) = (3 + 1) / 2 = 2
x_{2} = (- b - √D) / (2 a) = (3 - 1) / 2 = 1

b)
Given
x ^{2} / 2 = - 8 - 4 x
Multiply all terms in the above equation by 2, to eliminate the denominator, simplify and write it in standard form.
x ^{2} + 8 x + 16 = 0
Identify the coefficients a,b and c.
a = 1, b = 8 , c = 16
Find the discriminant D
D = b^{2} - 4 a c = 8^{2} - 4 (1)(16) = 64 - 64 = 0.
Since the discriminant is equal to zero, the above equation has one real solution.
x = - b / 2 a = -8/2 = -4

c)
Given
x ^{2} - 4x + 5 = 0
Identify the coefficients a,b and c.
a = 1, b = - 4 , c = 5
Find the discriminant D.
D = b^{2} - 4 a c = (-4)^{2} - 4 (1)(5) = 16 - 20 = -4.
Since the discriminant is negative, the above equation has two complex solutions.
x = (- b + √ D ) /(2 a) = (4 + √(-4) ) / 2 = (4 + 2 i) / 2 = 2 + i
x = (- b - √ D ) /(2 a) = (4 - √(-4) ) / 2 = (4 - 2 i) / 2 = 2 - i
where i = √(-1) is the imaginary unit.

2)
Given
x ^{2} + x + m + 1 = 0
Identify the coefficients a,b and c.
a = 1, b = 1 , c = m + 1
Find the discriminant D
D = b^{2} - 4ac = 1^{2} - 4(1)(m + 1) = -3 - 4m

a) For the equation to have one solution, the discriminant has to be equal to zero.
-3 - 4m = 0
Solve the above equation for m.
m = -3/4

b) For the equation to have 2 real solutions, the discriminant has to be greater than zero.
-3 - 4m > 0
The inequality -3 - 4m > 0 has the following solution set.
(-∞ , - 3/4)

c) For the equation to have 2 complex solutions, the discriminant has to be less than zero.
-3 - 4m < 0
The inequality -3 - 4m > 0 has the following solution set.
(- 3/4 , + ∞)