To solve the problem of subtracting the expression [tex]\(3x^2 - y^2\)[/tex] from [tex]\(5x^2 - 2y + 3y^2\)[/tex], we follow these steps:
1. Start with the given expressions:
- Expression 1: [tex]\(3x^2 - y^2\)[/tex]
- Expression 2: [tex]\(5x^2 - 2y + 3y^2\)[/tex]
2. Set up the subtraction:
To subtract [tex]\(3x^2 - y^2\)[/tex] from [tex]\(5x^2 - 2y + 3y^2\)[/tex], we write it as:
[tex]\[
(5x^2 - 2y + 3y^2) - (3x^2 - y^2)
\][/tex]
3. Distribute the negative sign:
When we subtract the second expression, the negative sign distributes to each term in the expression [tex]\(3x^2 - y^2\)[/tex]:
[tex]\[
5x^2 - 2y + 3y^2 - 3x^2 + y^2
\][/tex]
4. Combine like terms:
Now, we combine the like terms in the resulting expression:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(5x^2 - 3x^2 = 2x^2\)[/tex]
- Combine the [tex]\(y^2\)[/tex] terms: [tex]\(3y^2 + y^2 = 4y^2\)[/tex]
- The [tex]\(-2y\)[/tex] term remains unchanged
Thus, the final expression is:
[tex]\[
2x^2 + 4y^2 - 2y
\][/tex]
Therefore, the result of subtracting [tex]\(3x^2 - y^2\)[/tex] from [tex]\(5x^2 - 2y + 3y^2\)[/tex] is:
[tex]\[
2x^2 + 4y^2 - 2y
\][/tex]
