Step-by-step explanation:

1. completing the square

x² + 10x + ... = (x + a)² = x² + 2ax + a²

10 = 2a

a = 5

so,

x² + 10x + 10 = -22

is

(x + 5)² - 15 = -22

(x + 5)² = -7

x + 5 = i×sqrt(7)

the solutions are

x = i×sqrt(7) - 5

x = -i×sqrt(7) - 5

2. is easy via factoring

that means there has to be an easy way to define

x² + 3x - 40 = (x + a)(x + b)

remember,

(x + a)(x + b) = x² + (a+b)x + ab

since the constant term is -40, we know either a or b had to be negative (and the other positive).

a × b = -40

so, what factors deliver 40 ?

1×40

2×20

4×10

5×8

the x-term factor is 3.

so,

a + b = 3.

therefore, it must be the combination 8-5.

so, the factoring is

(x + 8)(x - 5) = 0

the solutions are therefore

x = -8

x = 5

3. quadratic formula

x² + 3x + 1 = 0

a = 1

b = 3

c = 1

x = (-b ± sqrt(b² - 4ac))/(2a)

x = (-3 ± sqrt(3² - 4×1×1))/(2×1) = (-3 ± sqrt(9 - 4))/2 =

= (-3 ± sqrt(5))/2

the solutions are

x = (-3 + sqrt(5))/2

x = (-3 - sqrt(5))/2

therefore

4. 2 non-real solutions (called also complex or irregular)

5. 2 rational solutions

6. 2 irrational solutions