1. **State the problem:** We need to find the positive solution to the quadratic equation $$7x^2 - 20x - 32 = 0.$$\n\n2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0,$$ the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$\n\n3. **Identify coefficients:** Here, $$a = 7,$$ $$b = -20,$$ and $$c = -32.$$\n\n4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-20)^2 - 4 \times 7 \times (-32) = 400 + 896 = 1296.$$\n\n5. **Find the square root of the discriminant:** $$\sqrt{1296} = 36.$$\n\n6. **Apply the quadratic formula:** $$x = \frac{-(-20) \pm 36}{2 \times 7} = \frac{20 \pm 36}{14}.$$\n\n7. **Calculate the two solutions:**\n\n- $$x_1 = \frac{20 + 36}{14} = \frac{56}{14} = 4.$$\n- $$x_2 = \frac{20 - 36}{14} = \frac{\cancel{20} - \cancel{36}}{\cancel{14}} = \frac{-16}{14} = -\frac{8}{7}.$$\n\n8. **Select the positive solution:** The positive solution is $$\boxed{4}.$$