The first problem is to simplify the expression $(y - \frac{2}{y+1}) \div (1 - \frac{2}{y+1})$. The second problem is to determine the number of sides of a regular polygon with an interior angle of 150 degrees.

1. Problem Description

The first problem is to simplify the expression $(y - \frac{2}{y+1}) \div (1 - \frac{2}{y+1})$.

The second problem is to determine the number of sides of a regular polygon with an interior angle of 150 degrees.

2. Solution Steps

Problem 23: Simplify $(y - \frac{2}{y+1}) \div (1 - \frac{2}{y+1})$

First, simplify the expression in the first set of parentheses:

$y - \frac{2}{y+1} = \frac{y(y+1)}{y+1} - \frac{2}{y+1} = \frac{y^2+y-2}{y+1}$

Next, simplify the expression in the second set of parentheses:

$1 - \frac{2}{y+1} = \frac{y+1}{y+1} - \frac{2}{y+1} = \frac{y+1-2}{y+1} = \frac{y-1}{y+1}$

Now, divide the first expression by the second expression:

$\frac{y^2+y-2}{y+1} \div \frac{y-1}{y+1} = \frac{y^2+y-2}{y+1} \times \frac{y+1}{y-1}$

$= \frac{(y^2+y-2)(y+1)}{(y+1)(y-1)}$

Since $y^2+y-2 = (y+2)(y-1)$, we have:

$= \frac{(y+2)(y-1)(y+1)}{(y+1)(y-1)}$

$= y+2$

Problem 24: Find the number of sides of a regular polygon with interior angle 150 degrees.

The formula for the interior angle of a regular n-sided polygon is:

$Interior Angle = \frac{(n-2) \times 180}{n}$

We are given that the interior angle is 150 degrees. So,

$150 = \frac{(n-2) \times 180}{n}$

$150n = 180n - 360$

$30n = 360$

$n = \frac{360}{30}$

$n = 12$

3. Final Answer

Problem 23: D. $y+2$

Problem 24: C. 12