Question

multiple attempts this test a... force completion this test can be saved and resumed at any... your answers are saved automatically. remaining time: 1 hour, 17 minutes, 49 seconds. question completion status: ! moving to another question will save this response. question 13 expand the following algebraic expression using the distributive prop ( b(b^2 + 4b - 2) + b^2(b + 8) + 2(b + 1) ) ( \bigcirc 2b^3 + 12b^2 - 4b + 1 ) ( \bigcirc 2b^3 + 12b^2 + 1 ) ( \bigcirc \text{nothing in this list is correct.} ) ( \bigcirc b^3 + 2b^2 + 11b + 1 ) ( \bigcirc b^3 + 2b^2 + 3b - 4 )

Explanation:

Step1: Expand each term

First, expand \( b(b^{2}+4b - 2) \):
Using the distributive property \( a(b + c + d)=ab+ac + ad \), we get \( b\times b^{2}+b\times4b - b\times2 = b^{3}+4b^{2}-2b \).

Next, expand \( b^{2}(b + 8) \):
Using the distributive property, we get \( b^{2}\times b + b^{2}\times8 = b^{3}+8b^{2} \).

Then, expand \( 2(b + 1) \):
Using the distributive property, we get \( 2\times b+2\times1 = 2b + 2 \).

Step2: Combine like terms

Now, add all the expanded terms together:
\( (b^{3}+4b^{2}-2b)+(b^{3}+8b^{2})+(2b + 2) \)
Combine the \( b^{3} \) terms: \( b^{3}+b^{3}=2b^{3} \)
Combine the \( b^{2} \) terms: \( 4b^{2}+8b^{2}=12b^{2} \)
Combine the \( b \) terms: \( -2b + 2b = 0 \)
Combine the constant terms: \( 2 \) (Wait, there is a mistake here. Wait, let's re - check the expansion of \( 2(b + 1) \) is \( 2b+2 \), and the \( b \) terms from \( b(b^{2}+4b - 2) \) is \( -2b \), from \( b^{2}(b + 8) \) there is no \( b \) term, and from \( 2(b + 1) \) is \( 2b \). So \( -2b+2b = 0 \), and the constant term is \( 2 \)? But the options do not have a constant term of 2. Wait, let's re - do the expansion:

Wait, the original expression is \( b(b^{2}+4b - 2)+b^{2}(b + 8)+2(b + 1) \)

Expanding \( b(b^{2}+4b - 2)=b^{3}+4b^{2}-2b \)

Expanding \( b^{2}(b + 8)=b^{3}+8b^{2} \)

Expanding \( 2(b + 1)=2b + 2 \)

Now, sum them up:

\( b^{3}+4b^{2}-2b + b^{3}+8b^{2}+2b + 2 \)

Now, combine like terms:

• For \( b^{3} \): \( 1b^{3}+1b^{3}=2b^{3} \)
• For \( b^{2} \): \( 4b^{2}+8b^{2}=12b^{2} \)
• For \( b \): \( -2b + 2b = 0 \)
• For constants: \( 2 \)

So the expanded form is \( 2b^{3}+12b^{2}+2 \). But none of the given options match this. Wait, maybe I made a mistake in the problem statement? Wait, the options are:

1. \( 2b^{3}+12b^{2}-4b + 1 \)
2. \( 2b^{3}+12b^{2}+1 \)
3. Nothing in this list is correct.
4. \( b^{3}+2b^{2}+11b + 1 \)
5. \( b^{3}+2b^{2}+3b - 4 \)

Since our result is \( 2b^{3}+12b^{2}+2 \), which is not in the list, the correct option is "Nothing in this list is correct."

Answer:

Nothing in this list is correct.