To determine which expressions are equivalent to [tex]\(2(4f + 2g)\)[/tex], we first need to simplify this expression.
Starting with:
[tex]\[ 2(4f + 2g) \][/tex]
Let's distribute the 2 to both terms inside the parentheses:
[tex]\[ 2 \cdot 4f + 2 \cdot 2g \][/tex]
This simplifies to:
[tex]\[ 8f + 4g \][/tex]
Now we need to evaluate each given option to see if they are equivalent to [tex]\(8f + 4g\)[/tex].
(A) [tex]\(8f + 2g\)[/tex]
This is not equivalent to [tex]\(8f + 4g\)[/tex] because the coefficient of [tex]\(g\)[/tex] is different.
(B) [tex]\(2f(4 + 2g)\)[/tex]
To simplify this, distribute [tex]\(2f\)[/tex]:
[tex]\[ 2f \cdot 4 + 2f \cdot 2g = 8f + 4fg \][/tex]
This expression involves [tex]\(4fg\)[/tex], which is not equivalent to [tex]\(8f + 4g\)[/tex].
(C) [tex]\(8f + 4g\)[/tex]
This is exactly the same as our simplified expression [tex]\(8f + 4g\)[/tex].
(D) [tex]\(4(2f + g)\)[/tex]
Let’s distribute the 4 in this expression:
[tex]\[ 4 \cdot 2f + 4 \cdot g = 8f + 4g \][/tex]
This is equivalent to [tex]\(8f + 4g\)[/tex].
(E) [tex]\(4f + 4f + 4g\)[/tex]
Combine like terms:
[tex]\[ 4f + 4f = 8f \][/tex]
[tex]\[ 8f + 4g \][/tex]
This is also equivalent to [tex]\(8f + 4g\)[/tex].
Thus, the equivalent expressions to [tex]\(2(4f + 2g)\)[/tex] are:
[tex]\[ C) 8f + 4g \][/tex]
[tex]\[ D) 4(2f + g) \][/tex]
[tex]\[ E) 4f + 4f + 4g \][/tex]
