To determine which expressions are equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex], we will substitute [tex]\( x = -1 \)[/tex] into each expression and simplify.
1. For the expression [tex]\(\frac{4(x+1)}{4x+5}\)[/tex]:
[tex]\[
\frac{4(-1+1)}{4(-1)+5} = \frac{4(0)}{-4+5} = \frac{0}{1} = 0
\][/tex]
So, [tex]\(\frac{4(x+1)}{4x+5}\)[/tex] is equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex].
2. For the expression [tex]\(\frac{4(x-1)}{5-4x}\)[/tex]:
[tex]\[
\frac{4(-1-1)}{5-4(-1)} = \frac{4(-2)}{5+4} = \frac{-8}{9} \neq 0
\][/tex]
So, [tex]\(\frac{4(x-1)}{5-4x}\)[/tex] is not equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex].
3. For the expression [tex]\(\frac{4(x-(-1))}{4x+5}\)[/tex]:
[tex]\[
\frac{4(-1-(-1))}{4(-1)+5} = \frac{4(-1+1)}{-4+5} = \frac{4(0)}{1} = 0
\][/tex]
So, [tex]\(\frac{4(x-(-1))}{4x+5}\)[/tex] is equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex].
4. For the expression [tex]\(\frac{4(x+(-1))}{4x+5}\)[/tex]:
[tex]\[
\frac{4(-1+(-1))}{4(-1)+5} = \frac{4(-2)}{-4+5} = \frac{-8}{1} = -8 \neq 0
\][/tex]
So, [tex]\(\frac{4(x+(-1))}{4x+5}\)[/tex] is not equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex].
5. For the expression [tex]\(\frac{4(x+1)}{5-4x}\)[/tex]:
[tex]\[
\frac{4(-1+1)}{5-4(-1)} = \frac{4(0)}{5+4} = \frac{0}{9} = 0
\][/tex]
So, [tex]\(\frac{4(x+1)}{5-4x}\)[/tex] is equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex].
Summarizing the results, the expressions that are equal to [tex]\(0\)[/tex] when [tex]\( x = -1 \)[/tex] are:
[tex]\[
\frac{4(x+1)}{4x+5}, \quad \frac{4(x-(-1))}{4x+5}, \quad \text{and} \quad \frac{4(x+1)}{5-4x}
\][/tex]
