Final answer:
The question involves simplifying a polynomial expression by applying the rules of exponents and polynomial multiplication. Specifically, we must expand and simplify the cubic term (2x²+5)³ and then multiply the result by the coefficient 4x outside the parentheses. The solution is achieved by applying the distributive property correctly.
Step-by-step explanation:
The student's question, which involves the substitution of a polynomial, falls under the category of Mathematics, specifically within the high school grade level. When dealing with the expression 4x(2x²+5)³, we aim to simplify the expression by following the rules of exponents and polynomial multiplication.
First, let's focus on the cubic term (2x²+5)³. According to the rules of cubing of exponentials, when we cube a polynomial, we raise each term in parentheses to the third power while taking care to apply the exponent to both the coefficient and the variable separately. In this case, the expression becomes (2³)(x²)³ + 3(2²)(x²)²(5) + 3(2)(x²)(5)² + (5)³. Simplifying further, we combine the constant terms and the variable terms with their corresponding exponents.
Finally, we multiply the simplified cubic term by the 4x that is outside the parentheses to obtain the full simplified polynomial, ensuring that we apply the distributive property correctly. Therefore, we perform 4x times each term in the expanded expression to achieve the final answer.