Let's simplify the expression step by step:

The given expression is:

$\left( \frac{5}{2} y^3 - 3y^2 + 8y - 45 \right) + \left( \frac{1}{2} y^3 + 4y^2 - 45 \right)$

First, let's group like terms:

• $y^3$ terms: $\frac{5}{2} y^3 + \frac{1}{2} y^3$
• $y^2$ terms: $-3y^2 + 4y^2$
• $y$ terms: $8y$ (no other $y$-terms to combine)
• Constant terms: $-45 - 45$

Now, simplify each group:

1. For the $y^3$ terms: $\frac{5}{2} y^3 + \frac{1}{2} y^3 = \frac{6}{2} y^3 = 3y^3$

2. For the $y^2$ terms: $-3y^2 + 4y^2 = 1y^2 = y^2$

3. The $y$-term is just $8y$.

4. For the constant terms: $-45 - 45 = -90$

The simplified expression is:

$3y^3 + y^2 + 8y - 90$

Let me know if you'd like further clarification or need more help!

Further questions:

1. How do you factor cubic polynomials like $3y^3 + y^2 + 8y - 90$?
2. What is the difference between polynomials and rational expressions?
3. Can this expression be factored? If so, how?
4. How do you find the roots of cubic polynomials?
5. What are the properties of polynomial functions, such as their behavior and graph shapes?

Tip:

When simplifying expressions, always group similar terms first, then combine them. This can prevent errors in more complicated expressions!