4. \( \frac{x}{x^{2}-6 x+9}+\frac{3}{2 x^{2}-5 x-3} \)

We start with the expression \[ \frac{x}{x^{2}-6x+9}+\frac{3}{2x^{2}-5x-3}\,. \] **Step 1. Factor the denominators.** Notice that \[ x^{2}-6x+9 = (x-3)^2\,. \] Next, factor \[ 2x^{2}-5x-3\,. \] We look for two numbers that multiply to \(2(-3)=-6\) and add to \(-5\). The numbers \(-6\) and \(1\) work, so we can factor by grouping: \[ 2x^{2}-5x-3 = 2x^{2}-6x+x-3 = 2x(x-3)+1(x-3) = (2x+1)(x-3)\,. \] Thus, the expression becomes \[ \frac{x}{(x-3)^2}+\frac{3}{(2x+1)(x-3)}\,. \] **Step 2. Find the common denominator.** The denominators are \((x-3)^2\) and \((2x+1)(x-3)\). The least common denominator (LCD) is \[ (2x+1)(x-3)^2\,. \] To have the same denominator the first fraction is multiplied by \((2x+1)\) (both numerator and denominator) and the second fraction by \((x-3)\): \[ \frac{x(2x+1)}{(2x+1)(x-3)^2}+\frac{3(x-3)}{(2x+1)(x-3)^2}\,. \] **Step 3. Combine the fractions.** Combine the numerators over the common denominator: \[ \frac{x(2x+1)+3(x-3)}{(2x+1)(x-3)^2}\,. \] **Step 4. Simplify the numerator.** Expand the numerator: \[ x(2x+1)=2x^2+x\,, \] \[ 3(x-3)=3x-9\,. \] So, \[ 2x^2+x+3x-9=2x^2+4x-9\,. \] Thus, the expression simplifies to \[ \frac{2x^2+4x-9}{(2x+1)(x-3)^2}\,. \] **Final Answer** \[ \frac{2x^2+4x-9}{(2x+1)(x-3)^2}\,. \]