1. **State the problem:** Solve the differential equation $$(-4x + 5y + 8) \, dx + (6x - 9y + 4) \, dy = 0.$$\n\n2. **Check if the equation is exact:** Let $M = -4x + 5y + 8$ and $N = 6x - 9y + 4$. Compute partial derivatives:\n$$\frac{\partial M}{\partial y} = 5, \quad \frac{\partial N}{\partial x} = 6.$$\nSince $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the equation is not exact.\n\n3. **Find an integrating factor:** Try an integrating factor depending on $x$ or $y$.\nCheck $\frac{\partial}{\partial y}\left(\frac{M}{N}\right)$ or $\frac{\partial}{\partial x}\left(\frac{N}{M}\right)$ to find a suitable integrating factor.\n\n4. **Try integrating factor depending on $x$:**\nCalculate $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 6 - 5 = 1$.\nCalculate $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \over M = \frac{1}{-4x + 5y + 8}$, which depends on both $x$ and $y$, so no simple integrating factor in $x$.\n\n5. **Try integrating factor depending on $y$:**\nCalculate $\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} = 5 - 6 = -1$.\nCalculate $\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} \over N = \frac{-1}{6x - 9y + 4}$, depends on both variables, so no simple integrating factor in $y$.\n\n6. **Try integrating factor depending on both variables or rearrange:**\nRewrite the equation as:\n$$(-4x + 5y + 8) + (6x - 9y + 4) \frac{dy}{dx} = 0,$$\nso\n$$\frac{dy}{dx} = \frac{4x - 5y - 8}{6x - 9y - 4}.$$\n\n7. **Attempt substitution:** Notice coefficients suggest substitution $v = \frac{y}{x}$. Then $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$.\n\n8. Substitute into the equation:\n$$v + x \frac{dv}{dx} = \frac{4x - 5(vx) - 8}{6x - 9(vx) - 4} = \frac{4x - 5vx - 8}{6x - 9vx - 4}.$$\nDivide numerator and denominator by $x$ (assuming $x \neq 0$):\n$$v + x \frac{dv}{dx} = \frac{4 - 5v - \frac{8}{x}}{6 - 9v - \frac{4}{x}}.$$\n\n9. This is complicated due to $\frac{8}{x}$ and $\frac{4}{x}$ terms, so substitution $v = y/x$ does not simplify easily.\n\n10. **Alternative approach:** Try integrating factor of form $\mu = e^{ax + by}$ or use advanced methods (not shown here).\n\n**Final note:** This differential equation is not exact and does not have a simple integrating factor depending only on $x$ or $y$. Further methods such as finding an integrating factor depending on both variables or using numerical methods may be required.