1. **State the problem:** Find the coordinates of the point that lies $\frac{3}{10}$ of the way from point $A(-5, -8)$ to point $B(12, 6)$.\n\n2. **Formula used:** The coordinates of a point dividing the segment from $A(x_1, y_1)$ to $B(x_2, y_2)$ in the ratio $t$ (where $t$ is the fraction of the way from $A$ to $B$) are given by:\n$$\left(x, y\right) = \left(x_1 + t(x_2 - x_1), y_1 + t(y_2 - y_1)\right)$$\n\n3. **Apply the formula:** Here, $t = \frac{3}{10}$, $x_1 = -5$, $y_1 = -8$, $x_2 = 12$, $y_2 = 6$.\nCalculate the $x$-coordinate:\n$$x = -5 + \frac{3}{10}(12 - (-5)) = -5 + \frac{3}{10}(17) = -5 + \frac{51}{10} = -5 + 5.1 = 0.1$$\nCalculate the $y$-coordinate:\n$$y = -8 + \frac{3}{10}(6 - (-8)) = -8 + \frac{3}{10}(14) = -8 + \frac{42}{10} = -8 + 4.2 = -3.8$$\n\n4. **Final answer:** The coordinates of the point $\frac{3}{10}$ of the way from $A$ to $B$ are $\boxed{(0.1, -3.8)}$.