Sure! Here's the detailed, step-by-step solution for expanding the expression [tex]\( -2 x^2 \left(3 x^2 - 4 x y + y^2\right) \)[/tex]:
1. Identify the original expression to expand:
[tex]\[
-2 x^2 \left(3 x^2 - 4 x y + y^2\right)
\][/tex]
2. Distribute [tex]\(-2x^2\)[/tex] to each term inside the parentheses:
[tex]\[
-2 x^2 \cdot 3 x^2 + (-2 x^2) \cdot (-4 x y) + (-2 x^2) \cdot y^2
\][/tex]
3. Multiply each pair individually:
- For the first term:
[tex]\[
-2 x^2 \cdot 3 x^2 = -6 x^4
\][/tex]
- For the second term:
[tex]\[
(-2 x^2) \cdot (-4 x y) = 8 x^3 y
\][/tex]
- For the third term:
[tex]\[
(-2 x^2) \cdot y^2 = -2 x^2 y^2
\][/tex]
4. Combine all the results from each multiplication:
[tex]\[
-6 x^4 + 8 x^3 y - 2 x^2 y^2
\][/tex]
So, after expanding the given expression step by step, we get the final expanded form:
[tex]\[
-6 x^4 + 8 x^3 y - 2 x^2 y^2
\][/tex]
