To determine which expression is equivalent to [tex]\((x+5)(2x^2 + 8)\)[/tex], we need to expand the given expression using the distributive property.
The distributive property states that [tex]\(a(b + c) = ab + ac\)[/tex]. Applying this property to our expression [tex]\((x+5)(2x^2 + 8)\)[/tex], we have:
[tex]\[
(x+5)(2x^2 + 8) = (x+5) \cdot 2x^2 + (x+5) \cdot 8
\][/tex]
Let's break this down step by step:
1. Distribute [tex]\( (x+5) \)[/tex] to [tex]\( 2x^2 \)[/tex]:
[tex]\[
(x+5) \cdot 2x^2 = 2x^2(x+5) = 2x^2 \cdot x + 2x^2 \cdot 5 = 2x^3 + 10x^2
\][/tex]
2. Distribute [tex]\( (x+5) \)[/tex] to [tex]\( 8 \)[/tex]:
[tex]\[
(x+5) \cdot 8 = 8(x+5) = 8x + 40
\][/tex]
So, combining these results:
[tex]\[
2x^3 + 10x^2 + 8x + 40
\][/tex]
Now, let's examine the choices given:
A. [tex]\((x+5)(2x) + (x+5)(8)\)[/tex]
Expanding this:
[tex]\[
(x+5) \cdot 2x = 2x(x+5) = 2x^2+10x
\][/tex]
[tex]\[
(x+5) \cdot 8 = 8(x+5) = 8x + 40
\][/tex]
Combining these:
[tex]\[
2x^2 + 10x + 8x + 40 = 2x^2 + 18x + 40
\][/tex]
This is not the same as our expanded expression.
B. [tex]\((x+5)(2x^2) + (x+5)(8)\)[/tex]
Expanding this:
[tex]\[
(x+5) \cdot 2x^2 = 2x^2(x+5) = 2x^3 + 10x^2
\][/tex]
[tex]\[
(x+5) \cdot 8 = 8(x+5) = 8x + 40
\][/tex]
Combining these:
[tex]\[
2x^3 + 10x^2 + 8x + 40
\][/tex]
This matches our expanded expression.
C. [tex]\((x+5)(2x^2) \cdot (x+5)(8)\)[/tex]
This is not distributive. It means we are multiplying the two resultant quantities, which is not correct.
D. [tex]\((x+5)(2x) \cdot (x+5)(8)\)[/tex]
This is incorrect for the same reason as C; it's a multiplication of two terms rather than distribution.
The correct choice is:
B. [tex]\((x+5)(2x^2) + (x+5)(8)\)[/tex]
