Question

question 12
expand the following algebraic expression using the distribution property, then simplify:
$(b + 1)(b - 1) - (b^2 + 1)(b + 3)$
\bigcirc $b^2 + 2b - 3$
\bigcirc $-b^3 - 3b^2 - b - 4$
\bigcirc nothing in this list is correct.
\bigcirc $b^3 + 2b^2 + b + 4$
\bigcirc $-b^3 + 3b^2 - b - 4$

Explanation:

Step1: Expand \((b + 1)(b - 1)\)

Using the difference of squares formula \((x + y)(x - y)=x^{2}-y^{2}\), here \(x = b\), \(y = 1\), so \((b + 1)(b - 1)=b^{2}-1\).

Step2: Expand \((b^{2}+1)(b + 3)\)

Using the distributive property (FOIL for binomial times binomial, extended for polynomial times binomial):
\[ \begin{align*} (b^{2}+1)(b + 3)&=b^{2}\times b+b^{2}\times3+1\times b + 1\times3\\ &=b^{3}+3b^{2}+b + 3 \end{align*} \]

Step3: Subtract the two expanded expressions

We have \((b + 1)(b - 1)-(b^{2}+1)(b + 3)=(b^{2}-1)-(b^{3}+3b^{2}+b + 3)\)
Distribute the negative sign:
\[ \begin{align*} &b^{2}-1 - b^{3}-3b^{2}-b - 3\\ =&-b^{3}+(b^{2}-3b^{2})-b+(-1 - 3)\\ =&-b^{3}-2b^{2}-b - 4 \end{align*} \]
Wait, let's check the options. Wait, maybe I misread the original expression. Wait, the original expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\)? Wait, no, maybe the user's expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\)? Wait, let's re - expand:

Wait, maybe the original problem is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\). Let's do it again:

First, \((b + 1)(b - 1)=b^{2}-1\)

Second, \((b^{2}+1)(b + 3)=b^{3}+3b^{2}+b + 3\)

Then, \((b^{2}-1)-(b^{3}+3b^{2}+b + 3)=b^{2}-1 - b^{3}-3b^{2}-b - 3=-b^{3}-2b^{2}-b - 4\). But this is not in the options. Wait, maybe the original expression is \((b + 1)(b - 1)-(b^{2}-1)(b + 3)\)? No, the user wrote \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\). Wait, let's check the options again. The options are:

1. \(b^{2}+2b - 3\)

2. \(-b^{3}-3b^{2}-b - 4\) (Wait, the second option is \(-b^{3}-3b^{2}-b - 4\)? Wait, the user's second option is \(-b^{3}-3b^{2}-b - 4\)? Let's re - calculate:

Wait, \((b + 1)(b - 1)=b^{2}-1\)

\((b^{2}+1)(b + 3)=b^{3}+3b^{2}+b + 3\)

Subtracting: \(b^{2}-1-(b^{3}+3b^{2}+b + 3)=-b^{3}-2b^{2}-b - 4\). But the second option is \(-b^{3}-3b^{2}-b - 4\). Wait, maybe I made a mistake in the original expression. Wait, maybe the original expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\) or maybe \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\) has a typo. Wait, let's check the options again. The second option is \(-b^{3}-3b^{2}-b - 4\). Let's see:

If we expand \((b + 1)(b - 1)=b^{2}-1\)

\((b^{2}+1)(b + 3)=b^{3}+3b^{2}+b + 3\)

\(b^{2}-1 - b^{3}-3b^{2}-b - 3=-b^{3}-2b^{2}-b - 4\). Not matching. Wait, maybe the original expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\) is wrong, maybe it's \((b + 1)(b - 1)-(b^{2}-1)(b + 3)\)? Let's try:

\((b^{2}-1)(b + 3)=b^{3}+3b^{2}-b - 3\)

Then \((b^{2}-1)-(b^{3}+3b^{2}-b - 3)=b^{2}-1 - b^{3}-3b^{2}+b + 3=-b^{3}-2b^{2}+b + 2\). Not matching.

Wait, maybe the original problem is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\) and the second option is \(-b^{3}-3b^{2}-b - 4\). Let's check the expansion of \((b^{2}+1)(b + 3)\) again: \(b^{2}\times b=b^{3}\), \(b^{2}\times3 = 3b^{2}\), \(1\times b=b\), \(1\times3 = 3\). So that's correct. Then \((b^{2}-1)-(b^{3}+3b^{2}+b + 3)=-b^{3}-2b^{2}-b - 4\). The second option is \(-b^{3}-3b^{2}-b - 4\), which is close but not the same. Wait, maybe the original expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\) and the second option is a typo, or maybe I misread the exponent. Wait, the second option is \(-b^{3}-3b^{2}-b - 4\). Let's see, if we have \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\), and the second option is \(-b^{3}-3b^{2}-b - 4\), maybe the first term is \((b + 1)(b - 1)=b^{2}-1\), and the second term is \((b^{2}+1)(b + 3)=b^{3}+3b^{2}+b + 3\), so subtracting gives \(-b^{3}-2b^{2}-b - 4\). Since this is not in the options, but the second option is \(-b^{3}-3b^{2}-b - 4\), maybe there is a mistake in the problem. But according to the calculation, the expanded form is \(-b^{3}-2b^{2}-b - 4\), which is not in the options. Wait, the third option is "Nothing in this list is correct". But let's check again.

Wait, maybe the original expression is \((b + 1)(b - 1)-(b^{2}+1)(b + 3)\). Let's compute step by step:

1. Expand \((b + 1)(b - 1)\):
Using \((a + b)(a - b)=a^{2}-b^{2}\), so \((b + 1)(b - 1)=b^{2}-1\).

2. Expand \((b^{2}+1)(b + 3)\):
\[ \begin{align*} (b^{2}+1)(b + 3)&=b^{2}\times b + b^{2}\times3+1\times b+1\times3\\ &=b^{3}+3b^{2}+b + 3 \end{align*} \]

3. Subtract the two results:
\[ \begin{align*} (b^{2}-1)-(b^{3}+3b^{2}+b + 3)&=b^{2}-1 - b^{3}-3b^{2}-b - 3\\ &=-b^{3}+(b^{2}-3b^{2})-b+(-1 - 3)\\ &=-b^{3}-2b^{2}-b - 4 \end{align*} \]

Now, check the options:

- Option 1: \(b^{2}+2b - 3\) is wrong.

- Option 2: \(-b^{3}-3b^{2}-b - 4\) is wrong (our result has \(-2b^{2}\) instead of \(-3b^{2}\)).

- Option 3: "Nothing in this list is correct"

- Option 4: \(b^{3}+2b^{2}+b + 4\) is wrong.

- Option 5: \(-b^{3}+3b^{2}-b - 4\) is wrong.

So the correct option is the third one: "Nothing in this list is correct".

Answer:

Nothing in this list is correct