To simplify the expression given by:

$\frac{y + 1}{y - 1} - \frac{(y + 1)^2}{(y - 1)^2} + y^3 + 3y^2 + y + 3,$

we'll approach it step by step.

Step 1: Simplify the fractions

Let's start by simplifying the first two fractions:

1. First Fraction: $\frac{y + 1}{y - 1}$

   This fraction is already in its simplest form.

2. Second Fraction: $-\frac{(y + 1)^2}{(y - 1)^2}$

   This fraction is also in its simplest form, though it can be noted that it represents a square of the first fraction.

Step 2: Combine the fractions

Now, we need to combine the fractions over a common denominator:

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

To combine these, we rewrite the first fraction with the same denominator:

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

Simplify the numerator:

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

Expand $(y + 1)^2$:

$(y + 1)^2 = y^2 + 2y + 1$

So, the expression becomes:

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

Simplify the numerator:

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

Factor out the $-2$ in the numerator:

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

Step 3: Combine with the remaining terms

Now, add the polynomial $y^3 + 3y^2 + y + 3$ to this result:

$\frac{-2(y + 1)}{(y - 1)^2} + y^3 + 3y^2 + y + 3$

Step 4: Interpret the expression

We will rewrite the entire expression:

$y^3 + 3y^2 + y + 3 + \frac{-2(y + 1)}{(y - 1)^2}$

This expression includes a polynomial part and a rational function part.

Conclusion

The expression simplifies to:

$y^3 + 3y^2 + y + 3 + \frac{-2(y + 1)}{(y - 1)^2}$

If you want to find critical points or values where the expression simplifies further, let me know!

Would you like more details, or have any questions?

Follow-up Questions:

1. How can you find common denominators for combining fractions?
2. What techniques can be used to factor polynomials?
3. How do you determine the domain of a rational function?
4. How can polynomials be divided by other polynomials?
5. What are the steps to simplify complex rational expressions?

Tip: Always factor expressions completely and look for common denominators when combining fractions!