1. **State the problem:** Find the Cartesian equation of the line passing through the point $(-2,4,-5)$ and parallel to the line given by $$\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}.$$\n\n2. **Understand the given line:** The given line is in symmetric form where the direction ratios are the denominators: $3$, $5$, and $6$. So the direction vector of the line is $$\vec{d} = \langle 3, 5, 6 \rangle.$$\n\n3. **Parallel lines have the same direction vector:** The line we want is parallel to the given line, so it has the same direction vector $$\vec{d} = \langle 3, 5, 6 \rangle.$$\n\n4. **Use the point-direction form of a line in 3D:** A line passing through point $P_0(x_0,y_0,z_0)$ with direction vector $\vec{d} = \langle a,b,c \rangle$ can be written as parametric equations:\n$$x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct,$$\nwhere $t$ is a parameter.\n\n5. **Substitute the given point and direction vector:**\n$$x = -2 + 3t,$$\n$$y = 4 + 5t,$$\n$$z = -5 + 6t.$$\n\n6. **Convert parametric form to symmetric form:**\n$$\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}.$$\n\n7. **Write the Cartesian equations:** From the symmetric form, cross-multiply to eliminate the parameter and get two equations:\n$$5(x + 2) = 3(y - 4) \implies 5x + 10 = 3y - 12,$$\n$$6(x + 2) = 3(z + 5) \implies 6x + 12 = 3z + 15.$$\n\n8. **Simplify each equation:**\n$$5x - 3y = -22,$$\n$$6x - 3z = 3.$$\n\n**Final answer:** The Cartesian equations of the line are\n$$\boxed{5x - 3y = -22 \quad \text{and} \quad 6x - 3z = 3}.$$