Let's solve the expression step by step:

Given Expression:

$\frac{y + 4}{y - 7} + \frac{y^2 + 9y + 20}{y^2 - 2y - 35} - \frac{y^2 - 3y - 28}{y^2 - 14y + 49}$

Step 1: Factor the Denominators and Numerators

First, we'll factor all the quadratic expressions in both the numerators and denominators.

1. Factor the first denominator $y - 7$:

   • No further factoring needed.

2. Factor the second denominator $y^2 - 2y - 35$:

   • We look for two numbers that multiply to $-35$ and add to $-2$.
   • These numbers are $-7$ and $5$, so the factorization is: $y^2 - 2y - 35 = (y - 7)(y + 5)$

3. Factor the third denominator $y^2 - 14y + 49$:

   • This is a perfect square trinomial, factoring to: $y^2 - 14y + 49 = (y - 7)^2$

Step 2: Rewrite the Expression with Factored Denominators

Now, the expression becomes: $\frac{y + 4}{y - 7} + \frac{y^2 + 9y + 20}{(y - 7)(y + 5)} - \frac{y^2 - 3y - 28}{(y - 7)^2}$

Step 3: Find a Common Denominator

The least common denominator (LCD) between the three fractions is $(y - 7)^2(y + 5)$.

• The first fraction needs to be multiplied by $(y - 7)(y + 5)$.
• The second fraction is already over $(y - 7)(y + 5)$.
• The third fraction needs to be multiplied by $y + 5$.

Step 4: Rewrite Each Fraction

Now, we rewrite each fraction with the common denominator.

1. First fraction: $\frac{y + 4}{y - 7} = \frac{(y + 4)(y - 7)(y + 5)}{(y - 7)^2(y + 5)}$

2. Second fraction: $\frac{y^2 + 9y + 20}{(y - 7)(y + 5)} = \frac{(y^2 + 9y + 20)(y - 7)}{(y - 7)^2(y + 5)}$

3. Third fraction: $\frac{y^2 - 3y - 28}{(y - 7)^2} = \frac{(y^2 - 3y - 28)(y + 5)}{(y - 7)^2(y + 5)}$

Step 5: Combine the Fractions

Now, add and subtract the numerators over the common denominator $(y - 7)^2(y + 5)$.

$\frac{(y + 4)(y - 7)(y + 5) + (y^2 + 9y + 20)(y - 7) - (y^2 - 3y - 28)(y + 5)}{(y - 7)^2(y + 5)}$

At this point, we can simplify the numerators. Let's expand each term:

1. Expand $(y + 4)(y - 7)(y + 5)$: = y^3 + 4y^2 - 2y^2 - 8y - 35y - 140 = y^3 + 2y^2 - 43y - 140$$
2. Expand $(y^2 + 9y + 20)(y - 7)$: = y^3 + 2y^2 - 43y - 140$$
3. Expand $(y^2 - 3y - 28)(y + 5)$: = y^3 + 2y^2 - 43y - 140$$

Step 6: Combine and Simplify

Now we can combine the numerators:

$(y^3 + 2y^2 - 43y - 140) + (y^3 + 2y^2 - 43y - 140) - (y^3 + 2y^2 - 43y - 140)$

Simplifying the expression:

$y^3 + 2y^2 - 43y - 140 + y^3 + 2y^2 - 43y - 140 - y^3 - 2y^2 + 43y + 140$

The terms cancel out:

$= 0$

Final Answer:

The result of the expression is: $\boxed{0}$

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Would you like to see more detailed steps or a deeper explanation of any part of the solution?

Additional Questions:

1. How do you factor quadratic expressions like $y^2 - 2y - 35$?
2. What is the process for finding the least common denominator of rational expressions?
3. Can you explain how to combine rational expressions with different denominators?
4. Why is factoring useful when simplifying rational expressions?
5. How can you check if your simplified expression is correct?

Tip: Always check for common factors in the numerator and denominator to simplify further if possible!