To simplify the expression [tex]\(\left(-6 y^2 + 4 y - 3 x\right) - \left(4 y^2 - 5 y + 6 x\right)\)[/tex], follow these steps:
1. Distribute the negative sign to each term inside the parentheses on the right:
[tex]\[
\left(-6 y^2 + 4 y - 3 x\right) - \left(4 y^2 - 5 y + 6 x\right) = \left(-6 y^2 + 4 y - 3 x\right) - 4 y^2 + 5 y - 6 x
\][/tex]
2. Simplify by combining like terms:
- Combine the [tex]\(y^2\)[/tex] terms: [tex]\(-6 y^2 - 4 y^2\)[/tex]
- Combine the [tex]\(y\)[/tex] terms: [tex]\(4 y + 5 y\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-3 x - 6 x\)[/tex]
3. Perform the arithmetic operations for each type of term:
- For the [tex]\(y^2\)[/tex] terms: [tex]\(-6 y^2 - 4 y^2 = -10 y^2\)[/tex]
- For the [tex]\(y\)[/tex] terms: [tex]\(4 y + 5 y = 9 y\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-3 x - 6 x = -9 x\)[/tex]
4. Combine these results to form the final simplified expression:
[tex]\[
\left(-10 y^2 + 9 y - 9 x\right)
\][/tex]
So, the simplified expression is [tex]\(-10 y^2 + 9 y - 9 x\)[/tex].
The correct answer from the given options is:
[tex]\[
-10 y^2 + 9 y - 9 x
\][/tex]
