$\frac{{2x}}{{2{x^2} - 5x + 3}} + \frac{{13x}}{{2{x^2} + x + 3}} = 6$ (1)Điều kiện: $x \ne 1,x \ne \frac{3}{2}$$(1) \Leftrightarrow 4 - \frac{{2x}}{{2{x^2} - 5x + 3}} + 2 - \frac{{13x}}{{2{x^2} + x + 3}} = 0$$\Leftrightarrow \frac{{8{x^2} - 22x + 12}}{{2{x^2} - 5x + 3}} + \frac{{4{x^2} - 11x + 6}}{{2{x^2} + x + 3}} = 0$$\Leftrightarrow (4{x^2} - 11x + 6)(\frac{2}{{2{x^2} - 5x + 3}} + \frac{1}{{2{x^2} + x + 3}}) = 0$$\Leftrightarrow 4{x^2} - 11x + 6 = 0$ (2) hay $2(2{x^2} + x + 3) + 2{x^2} - 5x + 3 = 0$ (3)$(2) \Leftrightarrow x = \frac{3}{4} \vee x = 2$$(3) \Leftrightarrow 2(2{x^2} + x + 3) + 2{x^2} - 5x + 3 = 0 \Leftrightarrow 6{x^2} - 3x + 9 = 0 (VN)$Vậy phương trình có hai nghiệm phân biệt$x = \frac{3}{4}$hay$ x=2$.

$\frac{{2x}}{{2{x^2} - 5x + 3}} + \frac{{13x}}{{2{x^2} + x + 3}} = 6$ (1)Điều kiện: $x \ne 1,x \ne \frac{3}{2}$$(1) \Leftrightarrow 4 - \frac{{2x}}{{2{x^2} - 5x + 3}} + 2 - \frac{{13x}}{{2{x^2} + x + 3}} = 0$$\Leftrightarrow \frac{{8{x^2} - 22x + 12}}{{2{x^2} - 5x + 3}} + \frac{{4{x^2} - 11x + 6}}{{2{x^2} + x + 3}} = 0$$\Leftrightarrow (4{x^2} - 11x + 6)(\frac{2}{{2{x^2} - 5x + 3}} + \frac{1}{{2{x^2} + x + 3}}) = 0$$\Leftrightarrow 4{x^2} - 11x + 6 = 0$ (2) hay $2(2{x^2} + x + 3) + 2{x^2} - 5x + 3 = 0$ (3)$(2) \Leftrightarrow x = \frac{3}{4} \vee x = 2$$(3) \Leftrightarrow 6{x^2} - 3x + 9 = 0 (VN)$Vậy phương trình có hai nghiệm phân biệt $x = \frac{3}{4}$ hay $x=2$.