To simplify the expression [tex]\(\frac{x+3}{x^3-2 x-3} \div \frac{z^2+2 x-3}{x+1}\)[/tex], we need to convert the division into a multiplication by the reciprocal:
[tex]\[
\frac{x+3}{x^3-2 x-3} \div \frac{z^2+2 x-3}{x+1} = \frac{x+3}{x^3-2 x-3} \times \frac{x+1}{z^2+2 x-3}
\][/tex]
Next, we multiply the two fractions:
[tex]\[
\frac{(x+3)(x+1)}{(x^3-2 x-3)(z^2+2 x-3)}
\][/tex]
Now, we need to simplify this expression. After simplification, it becomes:
[tex]\[
-(x + 1)(x + 3) / ((2x + z^2 - 3)(-x^3 + 2x + 3))
\][/tex]
Since the simplified result is:
[tex]\[
\frac{-(x + 1)(x + 3)}{(2 x + z^2 - 3)(-x^3 + 2 x + 3)},
\][/tex]
and it does not match any of the given choices exactly, but we can compare the form of the given expressions.
Given the choices:
A. [tex]\(\frac{1}{x^2-2 z-3}\)[/tex]
B. [tex]\(\frac{1}{x^1-4 x+3}\)[/tex]
C. [tex]\(\frac{1}{x^1+2 x-3}\)[/tex]
D. [tex]\(\frac{x+3}{z+1}\)[/tex]
None of these given options exactly match the derived and simplified expression.
Based on your given choices and correct simplification, we conclude that none of the choices A, B, C, or D are correct. The simplified expression involving more complex elements rooted in the answer [tex]\(-(x + 1)(x + 3)/((2x + z2 - 3)(-x3 + 2x + 3))\)[/tex], does not fit clearly into any provided option. Therefore, none of the above answers appear valid as per the context given.
