To simplify the given expression [tex]\(3\left[2\left(-y^2 + y\right) - 3\right] - 3(2x + y)\)[/tex], we can follow a step-by-step process:
1. Distribute the constant inside the first set of brackets:
First, consider the inner expression: [tex]\(2(-y^2 + y)\)[/tex].
[tex]\[
2(-y^2 + y) = 2 \cdot (-y^2) + 2 \cdot y = -2y^2 + 2y
\][/tex]
2. Simplify the expression inside the brackets:
Substitute back into the expression within the brackets:
[tex]\[
2(-y^2 + y) - 3 = -2y^2 + 2y - 3
\][/tex]
3. Distribute the 3 across the simplified expression:
Now distribute the 3:
[tex]\[
3[-2y^2 + 2y - 3] = 3 \cdot (-2y^2) + 3 \cdot 2y + 3 \cdot (-3) = -6y^2 + 6y - 9
\][/tex]
4. Simplify the second part of the expression:
Distribute the [tex]\(-3\)[/tex] in the second part:
[tex]\[
-3(2x + y) = -3 \cdot 2x - 3 \cdot y = -6x - 3y
\][/tex]
5. Combine the simplified parts:
Now add the two parts together:
[tex]\[
-6y^2 + 6y - 9 - 6x - 3y
\][/tex]
6. Combine like terms:
Combine the [tex]\(y\)[/tex] terms:
[tex]\[
6y - 3y = 3y
\][/tex]
Thus, the final simplified expression is:
[tex]\[
-6x - 6y^2 + 3y - 9
\][/tex]
Therefore, the fully simplified expression is:
[tex]\[
-6x - 6y^2 + 3y - 9
\][/tex]
