To solve the given expression [tex]\((2a) \left(3x^2 + 2xy - 4y^3 \right)\)[/tex], we will break it down into smaller steps.
### Expression Simplification
The given expression is:
[tex]\[ (2a) \left(3x^2 + 2xy - 4y^3 \right) \][/tex]
This indicates we need to multiply the term [tex]\(2a\)[/tex] by each term inside the parentheses. Let's do this step-by-step:
1. Multiply [tex]\(2a\)[/tex] by the first term inside the parentheses:
[tex]\[ 2a \cdot 3x^2 = 6ax^2 \][/tex]
2. Multiply [tex]\(2a\)[/tex] by the second term inside the parentheses:
[tex]\[ 2a \cdot 2xy = 4axy \][/tex]
3. Multiply [tex]\(2a\)[/tex] by the third term inside the parentheses:
[tex]\[ 2a \cdot (-4y^3) = -8ay^3 \][/tex]
### Combine the Terms
After performing the multiplication, we combine the results:
[tex]\[ 6ax^2 + 4axy - 8ay^3 \][/tex]
Thus, the full simplified expression is:
[tex]\[ \boxed{2a(3x^2 + 2xy - 4y^3) = 6ax^2 + 4axy - 8ay^3} \][/tex]
This is the fully simplified form of the given mathematical expression.
