Final answer:
To factor the expression 3x⁴y³ - 48y³ completely, we can first factor out the common term y³, then factor out the common factor 3 inside the parentheses. Finally, we can factor the expression (x² - 4) as the difference of squares.
Step-by-step explanation:
To factor the expression 3x⁴y³ - 48y³, we can first factor out the common term, which is y³. This gives us y³(3x⁴ - 48).
Next, we can factor out the common factor of 3 from the terms inside the parentheses. This gives us y³(3(x⁴ - 16)).
Now, we can further factor the expression inside the parentheses as the difference of squares: y³(3(x² - 4)(x² + 4)).
Finally, we can factor the expression (x² - 4) as the difference of squares, giving us y³(3(x - 2)(x + 2))(x² + 4). Therefore, the completely factored form of the expression is y³(3(x - 2)(x + 2))(x² + 4).