Final answer:
Out of the given options, the completely factored form of the polynomial is option a) (8x²(x+3)²), as it can't be factored further and each term is expressed as a product of factors.
Step-by-step explanation:
To find the completely factored form of the polynomial, we must look for an expression that is broken down into factors that cannot be further simplified. In this case, the student has asked which of the following options represents the completely factored form of her polynomial: a) (8x²(x+3)²), b) (2x(4x²+5)), c) (2(4x²+5) (x+3)), or d) (2(4x²+5) (x+3²)).
Let's analyze each option separately:
- Option a) (8x²(x+3)²) seems to be in completely factored form because we have a coefficient 8x² multiplied by a binomial (x+3) raised to the second power, and there is no further factoring possible for the binomial.
- Option b) (2x(4x²+5)) is partially factored, but the trinomial 4x²+5 cannot be factored further without knowing additional constraints.
- Option c) (2(4x²+5) (x+3)) is not completely factored because the trinomial 4x²+5 is not factored, and neither is the binomial x+3.
- Option d) (2(4x²+5) (x+3²)) is incorrect due to the presence of (x+3²), which suggests that +3² is a term on its own, which does not make sense mathematically as '3²' should apply to the whole term within the parenthesis.
Therefore, the answer is option a) (8x²(x+3)²).