The mathematical expression shown in the image is:

$2y^2 - 2y - 3y + 3x^2$

To simplify this expression, we can combine like terms:

1. Combine the $y$ terms: $-2y - 3y = -5y$.
2. There are no like terms for $2y^2$ and $3x^2$.

Thus, the simplified expression is:

$2y^2 - 5y + 3x^2$

Would you like more details on this simplification or have any further questions?

Here are five related questions that expand on this topic:

1. How do you factor quadratic expressions?
2. What are like terms in algebraic expressions?
3. Can you show how to complete the square for the expression $2y^2 - 5y + 3x^2$?
4. What are the applications of quadratic expressions in real-world scenarios?
5. How do you determine if a quadratic expression can be factored over the real numbers?

Tip: When simplifying expressions, always look for like terms to combine. This helps in reducing the complexity of the expression and makes it easier to work with.