1. **Problem statement:** Given the equation $$0 = \frac{(ع س + ن)( س - 1)( س + 4)}{س - 1}$$, find the value of the variable $ن$. 2. **Understanding the equation:** The expression is a rational function where the numerator is $(ع س + ن)( س - 1)( س + 4)$ and the denominator is $س - 1$. 3. **Simplify the expression:** Since $س - 1$ appears in both numerator and denominator, and assuming $س \neq 1$ to avoid division by zero, we can cancel $س - 1$: $$0 = \frac{(ع س + ن)\cancel{( س - 1)}( س + 4)}{\cancel{س - 1}} = (ع س + ن)( س + 4)$$ 4. **Set the simplified expression equal to zero:** $$0 = (ع س + ن)( س + 4)$$ 5. **Zero product property:** For the product to be zero, at least one factor must be zero: - $ع س + ن = 0$ - or $س + 4 = 0$ 6. **Find $ن$ when $س = -4$:** If $س = -4$, then $س + 4 = 0$, so the equation holds regardless of $ن$. 7. **Find $ن$ when $ع س + ن = 0$:** Rearranged: $$ن = -ع س$$ 8. **Conclusion:** The value of $ن$ depends on $ع$ and $س$. For the equation to hold for all $س$ except $س = 1$, $ن$ must satisfy: $$ن = -ع س$$ If you want a specific value, more information about $ع$ and $س$ is needed. **Final answer:** $$ن = -ع س$$