$\left(\frac{6x+1}{x^2-6x}+\frac{6x-1}{x^2+6x}\right)\cdot\frac{x^2-36}{x^2+1}$ $\frac{\left(\frac{x+1}{2x-1}-\frac{2x+5}{4x+4}\right)}{\left(\frac{x+1}{2x+5}-\frac{2x-1}{4x+4}\right)}$

Điều kiện xác định: \( x \neq 0; x \neq \pm 6; x \neq -2 \) Ta có: \[ \left(\frac{6x+1}{x^2-6x}+\frac{6x-1}{x^2+6x}\right)\cdot\frac{x^2-36}{x^2+1} \] \[ = \left[\frac{6x+1}{x(x-6)}+\frac{6x-1}{x(x+6)}\right]\cdot\frac{(x-6)(x+6)}{x^2+1} \] \[ = \left[\frac{(6x+1)(x+6)+(6x-1)(x-6)}{x(x-6)(x+6)}\right]\cdot\frac{(x-6)(x+6)}{x^2+1} \] \[ = \frac{(6x+1)(x+6)+(6x-1)(x-6)}{x(x^2+1)} \] \[ = \frac{6x^2+37x+6+6x^2-37x+6}{x(x^2+1)} \] \[ = \frac{12x^2+12}{x(x^2+1)} \] \[ = \frac{12(x^2+1)}{x(x^2+1)} \] \[ = \frac{12}{x} \] Tiếp theo, ta có: \[ \frac{\left(\frac{x+1}{2x-1}-\frac{2x+5}{4x+4}\right)}{\left(\frac{x+1}{2x+5}-\frac{2x-1}{4x+4}\right)} \] \[ = \frac{\left(\frac{x+1}{2x-1}-\frac{2x+5}{4(x+1)}\right)}{\left(\frac{x+1}{2x+5}-\frac{2x-1}{4(x+1)}\right)} \] \[ = \frac{\left(\frac{(x+1)^2-(2x+5)(2x-1)}{2(2x-1)(x+1)}\right)}{\left(\frac{(x+1)^2-(2x-1)(2x+5)}{4(2x+5)(x+1)}\right)} \] \[ = \frac{(x+1)^2-(2x+5)(2x-1)}{2(2x-1)(x+1)} \cdot \frac{4(2x+5)(x+1)}{(x+1)^2-(2x-1)(2x+5)} \] \[ = \frac{-2(2x^2+3x-3)}{2(2x-1)(x+1)} \cdot \frac{4(2x+5)(x+1)}{-2(2x^2+3x-3)} \] \[ = \frac{4(2x+5)}{2(2x-1)} \] \[ = \frac{2(2x+5)}{2x-1} \] Cuối cùng, ta có: \[ \frac{12}{x} \div \frac{2(2x+5)}{2x-1} \] \[ = \frac{12}{x} \cdot \frac{2x-1}{2(2x+5)} \] \[ = \frac{12(2x-1)}{2x(2x+5)} \] \[ = \frac{6(2x-1)}{x(2x+5)} \]