To find the rational expression equivalent to [tex]\(\frac{4}{a-3}\)[/tex], we need to consider each option given and verify whether it simplifies to the original expression.
First, let's review the operations involved for each option:
Option A: [tex]\(\frac{x+2}{x-3}-\frac{4}{x+2}\)[/tex]
Since option A involves subtraction, we need common denominators to combine these fractions. After finding the common denominator and combining the fractions, simplify the resulting expression.
Option B: [tex]\(\frac{z+2}{x-3}+\frac{4}{x+2}\)[/tex]
In option B, the numerators being added directly, first, we’d need common denominators. Combining or simplifying the resulting fraction might give us the form we seek.
Option C: [tex]\(\frac{x+2}{x-3} \cdot \frac{4}{x+2}\)[/tex]
Option C involves multiplication of fractions. When multiplying fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{(x+2) \cdot 4}{(x-3) \cdot (x+2)}
\][/tex]
Notice the [tex]\(x+2\)[/tex] in the numerator and denominator cancel out:
[tex]\[
\frac{4}{x-3}
\][/tex]
Option D: [tex]\(\frac{x+2}{x-3} \div \frac{4}{x+2}\)[/tex]
Option D involves division of fractions, which is the same as multiplying by the reciprocal of the divisor:
[tex]\[
\frac{x+2}{x-3} \div \frac{4}{x+2} = \frac{x+2}{x-3} \cdot \frac{x+2}{4}
\][/tex]
After rearranging:
[tex]\[
\frac{(x+2) \cdot (x+2)}{(x-3) \cdot 4}
\][/tex]
This does not simplify to [tex]\(\frac{4}{a-3}\)[/tex].
From the explanation, it is clear that Option C is exactly the expression that simplifies to the original [tex]\(\frac{4}{a-3}\)[/tex].
So, the correct option is:
C. [tex]\(\frac{x+2}{x-3} \cdot \frac{4}{x+2}\)[/tex]
