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 $N = 6x - 9y + 4$.\nCalculate $\frac{\partial M}{\partial y} = 5$ and $\frac{\partial N}{\partial x} = 6$. Since $5 \neq 6$, the equation is not exact.\n\n3. **Find an integrating factor:** Try an integrating factor depending on $x$ or $y$.\nCalculate $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 6 - 5 = 1$.\nCalculate $\frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} = \frac{1}{-4x + 5y + 8}$, which depends on both $x$ and $y$, so no simple integrating factor in $x$ alone.\nCalculate $\frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} = \frac{5 - 6}{6x - 9y + 4} = \frac{-1}{6x - 9y + 4}$, which also depends on both variables.\n\n4. **Try to find an integrating factor of the form $\mu = e^{ax+by}$:**\nWe want $\mu M$ and $\mu N$ to satisfy $\frac{\partial (\mu M)}{\partial y} = \frac{\partial (\mu N)}{\partial x}$.\nThis leads to the condition $a M + \frac{\partial M}{\partial y} = b N + \frac{\partial N}{\partial x}$.\nSubstitute values and solve for $a$ and $b$.\n\n5. **Alternatively, solve as a linear differential equation in $y$:**\nRewrite as $\frac{dy}{dx} = -\frac{M}{N} = -\frac{-4x + 5y + 8}{6x - 9y + 4} = \frac{4x - 5y - 8}{6x - 9y + 4}$.\n\n6. **This is a first-order ODE in $y$ and $x$.**\nRewrite as $$\frac{dy}{dx} = \frac{4x - 5y - 8}{6x - 9y + 4}.$$\n\n7. **Use substitution:** Let $v = y/x$, so $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$.\nSubstitute into the equation:\n$$v + x \frac{dv}{dx} = \frac{4x - 5vx - 8}{6x - 9vx + 4} = \frac{4 - 5v - \frac{8}{x}}{6 - 9v + \frac{4}{x}}.$$\n\n8. **For large $x$, approximate:**\n$$v + x \frac{dv}{dx} \approx \frac{4 - 5v}{6 - 9v}.$$\n\n9. **Solve the approximate separable equation:**\n$$x \frac{dv}{dx} = \frac{4 - 5v}{6 - 9v} - v = \frac{4 - 5v - v(6 - 9v)}{6 - 9v} = \frac{4 - 5v - 6v + 9v^2}{6 - 9v} = \frac{4 - 11v + 9v^2}{6 - 9v}.$$\n\n10. **Separate variables:**\n$$\frac{6 - 9v}{4 - 11v + 9v^2} dv = \frac{dx}{x}.$$\n\n11. **Integrate both sides:**\nThe integral on the left can be done by partial fractions; the right side integrates to $\ln|x| + C$.\n\n12. **Final implicit solution:**\nAfter integration and back-substitution, the solution is implicit in $x$ and $y$.\n\n**Answer:** The differential equation is not exact, and solving requires substitution and integration leading to an implicit solution involving $x$ and $y$.