Final answer:
The expression log2(8x^2 + 16x + 8) can be factored to get 3 + log2(x^2 + 2x + 1), as log2(8) simplifies to 3. The provided answer choices do not include this option, indicating a possible misunderstanding of the question or a mistake in the given options.
Step-by-step explanation:
The question asks us to expand the expression log2(8x^2 + 16x + 8). However, it seems there is a misunderstanding because the expression inside the logarithm is a quadratic, and simply expanding the logarithm function as written is not possible with the rules of logarithms. Logarithms can only be expanded by applying properties such as the product, quotient, and power properties when the argument of the logarithm can be expressed in terms of products, quotients, or powers. For log2(8x^2 + 16x + 8), we can first notice that all the terms in the quadratic are factors of 8, which gives us a clue on how to proceed.
First, factor out the greatest common factor, which is 8 in this case:
log2(8(x^2 + 2x + 1))
Then, apply the logarithm rule that allows us to move constants outside the log as a multiplier:
log2(8) + log2(x^2 + 2x + 1)
Now we recognize that 8 is 2 to the power of 3, so we can simplify further:
3 + log2(x^2 + 2x + 1)
Lastly, we can look to see if the quadratic can be factored or simplified, but in this case, the quadratic expression (x^2 + 2x + 1) is a perfect square trinomial (x+1)^2, yet without additional manipulation, we cannot simplify the expression further inside the log. The expansion of the expression, therefore, is option B) 3 + log2(x^2 + 2x + 1), though this option does not appear in the original list, suggesting an error in the provided options or a different intended question.