a) To solve the equation 2x + 4 = 2^3, we can start by simplifying the right side of the equation:

2^3 = 8

So the equation becomes:

2x + 4 = 8

Next, we can subtract 4 from both sides of the equation:

2x = 8 - 4
2x = 4

Finally, we divide both sides of the equation by 2 to solve for x:

x = 4/2
x = 2

Therefore, the solution to the equation is x = 2.

b) To solve the equation 3^x + 1 + 23 = 2.5^2.2022^0, we can start by simplifying the right side of the equation:

2.5^2 = 6.25
2022^0 = 1

So the equation becomes:

3^x + 1 + 23 = 6.25 * 1

Next, we can simplify the right side of the equation:

6.25 * 1 = 6.25

So the equation becomes:

3^x + 1 + 23 = 6.25

Next, we can subtract 1 and 23 from both sides of the equation:

3^x = 6.25 - 1 - 23
3^x = -17.75

Since the left side of the equation is a positive number (3^x), and the right side is a negative number (-17.75), there is no real solution to this equation.

c) To find the values of x that divide 180 and satisfy the condition 12 < x < 20, we can start by listing the factors of 180:

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

Next, we can identify the factors that are greater than 12 and less than 20:

15, 18

Therefore, the values of x that divide 180 and satisfy the condition 12 < x < 20 are x = 15 and x = 18.