5. \( \frac{x^{2}-x+2 x}{x^{2}-2 x-3} \div \frac{x^{2}-4 x+3}{1-x^{2}} \)

We start with the expression   [(x² – x + 2x) / (x² – 2x – 3)] ÷ [(x² – 4x + 3) / (1 – x²)]. Step 1. Simplify the numerator of the first fraction.   x² – x + 2x = x² + x = x(x + 1). Step 2. Factor the denominator of the first fraction.   x² – 2x – 3 factors as (x + 1)(x – 3). So the first fraction becomes   [x(x + 1)] / [(x + 1)(x – 3)]. Cancel the common factor (x + 1) (provided x ≠ –1):   = x / (x – 3). Step 3. Factor the numerator of the second fraction.   x² – 4x + 3 factors as (x – 1)(x – 3). Step 4. Factor the denominator of the second fraction.   1 – x² = –(x² – 1) = –(x – 1)(x + 1). So the second fraction becomes   [(x – 1)(x – 3)] / [– (x – 1)(x + 1)]. Cancel the common factor (x – 1) (provided x ≠ 1):   = (x – 3) / [– (x + 1)] = –(x – 3) / (x + 1). Step 5. Now we have   [x / (x – 3)] ÷ [–(x – 3) / (x + 1)]. Dividing by a fraction is the same as multiplying by its reciprocal:   = [x / (x – 3)] × [– (x + 1) / (x – 3)]   = – x (x + 1) / (x – 3)². Thus, the simplified expression is   – x(x + 1) / (x – 3)². Remember that during these cancellations, you must note the restrictions:  • x ≠ –1 (from canceling x + 1),  • x ≠ 1 (from canceling x – 1),  • x ≠ 3 (since x – 3 appears in a denominator). The final answer is   – x(x + 1) / (x – 3)², with x ≠ –1, 1, 3.