Sure, let's examine each option to determine which expression is equivalent to [tex]\(\frac{4}{z-3}\)[/tex].
### Option A:
[tex]\[
\frac{x-3}{x+2} \cdot \frac{x+2}{4}
\][/tex]
When we simplify this, we can cancel out [tex]\(\frac{x+2}{x+2}\)[/tex]:
[tex]\[
\frac{x-3}{x+2} \cdot \frac{x+2}{4} = \frac{(x-3) \cdot (x+2)}{(x+2) \cdot 4} = \frac{x-3}{4}
\][/tex]
This is not equivalent to [tex]\(\frac{4}{z-3}\)[/tex].
### Option B:
[tex]\[
\frac{x+2}{x-3} \cdot \frac{4}{x+2}
\][/tex]
When we simplify this, we can cancel out [tex]\(\frac{x+2}{x+2}\)[/tex]:
[tex]\[
\frac{x+2}{x-3} \cdot \frac{4}{x+2} = \frac{(x+2) \cdot 4}{(x-3) \cdot (x+2)} = \frac{4}{x-3}
\][/tex]
This simplifies correctly to [tex]\(\frac{4}{x-3}\)[/tex], which is equivalent to [tex]\(\frac{4}{z-3}\)[/tex] if we let [tex]\(x = z\)[/tex].
### Option C:
[tex]\[
\frac{x+2}{x-3} \div \frac{4}{x+2}
\][/tex]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{x+2}{x-3} \div \frac{4}{x+2} = \frac{x+2}{x-3} \cdot \frac{x+2}{4} = \frac{(x+2) \cdot (x+2)}{(x-3) \cdot 4} = \frac{(x+2)^2}{4(x-3)}
\][/tex]
This is not equivalent to [tex]\(\frac{4}{z-3}\)[/tex].
### Option D:
[tex]\[
\frac{x-3}{x+2} \div \frac{4}{z+2}
\][/tex]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{x-3}{x+2} \div \frac{4}{z+2} = \frac{x-3}{x+2} \cdot \frac{z+2}{4} = \frac{(x-3) \cdot (z+2)}{4(x+2)}
\][/tex]
This is not equivalent to [tex]\(\frac{4}{z-3}\)[/tex].
So, upon checking all the options, the correct equivalent expression to [tex]\(\frac{4}{z-3}\)[/tex] is:
[tex]\[
\boxed{B}
\][/tex]
