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question 13

expand the following algebraic expression using the distributive property, then simplify by gathering like terms.
( b(b^2 + 4b - 2) + b^2(b + 8) + 2(b + 1) )

( \bigcirc ) ( b^3 + 2b^2 + 3b - 4 )

( \bigcirc ) ( 2b^3 + 12b^2 - 4b + 1 )

( \bigcirc ) ( b^3 + 2b^2 + 11b + 1 )

( \bigcirc ) nothing in this list is correct.

( \bigcirc ) ( 2b^3 + 12b^2 + 1 )

Explanation:

Step1: Apply distributive property

First, distribute each term:

• For \( b(b^2 + 4b - 2) \), we get \( b \cdot b^2 + b \cdot 4b - b \cdot 2 = b^3 + 4b^2 - 2b \)
• For \( b^2(b + 8) \), we get \( b^2 \cdot b + b^2 \cdot 8 = b^3 + 8b^2 \)
• For \( 2(b + 1) \), we get \( 2 \cdot b + 2 \cdot 1 = 2b + 2 \)

Step2: Combine all expanded terms

Now, combine the three expanded expressions:
\( (b^3 + 4b^2 - 2b) + (b^3 + 8b^2) + (2b + 2) \)

Step3: Gather like terms

• For \( b^3 \) terms: \( b^3 + b^3 = 2b^3 \)
• For \( b^2 \) terms: \( 4b^2 + 8b^2 = 12b^2 \)
• For \( b \) terms: \( -2b + 2b = 0 \)
• For constant terms: \( 2 \)

Putting it all together, we have \( 2b^3 + 12b^2 + 2 \)

Wait, but let's check again. Wait, maybe I made a mistake. Wait, let's re - expand:

Wait, \( b(b^2 + 4b - 2)=b^3 + 4b^2-2b \)

\( b^2(b + 8)=b^3+8b^2 \)

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

Now, add them:

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

Now, combine like terms:

\( b^3 + b^3=2b^3 \)

\( 4b^2+8b^2 = 12b^2 \)

\( -2b+2b = 0 \)

\( 0 + 2=2 \)

So the simplified expression is \( 2b^3+12b^2 + 2 \)

Looking at the options, none of them match. Wait, maybe I made a mistake in the problem statement? Wait, the original expression is \( b(b^2 + 4b - 2)+b^2(b + 8)+2(b + 1) \)

Wait, let's check the options again. The options are:

1. \( b^3+2b^2+3b - 4 \)

1. \( 2b^3+12b^2-4b + 1 \)

1. \( b^3+2b^2+11b + 1 \)

1. Nothing in this list is correct.

1. \( 2b^3+12b^2 + 1 \)

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.