The expression in the image is:

$\frac{1}{y + 2} + \frac{9}{y + 1}$

To add these two rational expressions, we need to find a common denominator.

Step 1: Find the Least Common Denominator (LCD)

The denominators are $y + 2$ and $y + 1$. The least common denominator (LCD) will be the product of these two terms:

$\text{LCD} = (y + 2)(y + 1)$

Step 2: Rewrite Each Fraction with the LCD

Now, rewrite each fraction with the LCD as the denominator.

For $\frac{1}{y + 2}$, multiply both the numerator and the denominator by $y + 1$:

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

For $\frac{9}{y + 1}$, multiply both the numerator and the denominator by $y + 2$:

$\frac{9}{y + 1} = \frac{9 \cdot (y + 2)}{(y + 1)(y + 2)} = \frac{9(y + 2)}{(y + 1)(y + 2)}$

Step 3: Add the Fractions

Now, add the two fractions:

$\frac{y + 1}{(y + 2)(y + 1)} + \frac{9(y + 2)}{(y + 1)(y + 2)} = \frac{(y + 1) + 9(y + 2)}{(y + 1)(y + 2)}$

Step 4: Simplify the Numerator

Simplify the numerator:

$(y + 1) + 9(y + 2) = y + 1 + 9y + 18 = 10y + 19$

So, the expression becomes:

$\frac{10y + 19}{(y + 1)(y + 2)}$

Final Answer:

The simplified form of the given expression is:

$\frac{10y + 19}{(y + 1)(y + 2)}$

Let me know if you'd like further explanation or details!

Here are 5 related questions to expand on this concept:

1. How would you simplify an expression with three fractions with different denominators?
2. What happens if one of the terms in the denominator contains a square, like $y^2 + 1$?
3. How would you factor and simplify a rational expression like $\frac{y^2 + 3y + 2}{y^2 - 1}$?
4. How do you handle complex rational expressions where both the numerator and denominator are fractions?
5. Can you explain how to perform polynomial long division when simplifying rational expressions?

Tip: Always factor denominators when possible to find the least common denominator more easily!