1. **State the problem:** Find the equation of a circle that passes through the point $(5,4)$ and is concentric with the circle given by $$x^2 + y^2 - 8x + 6y + 20 = 0.$$\n\n2. **Recall the formula for a circle:** The general form of a circle's equation is $$x^2 + y^2 + 2gx + 2fy + c = 0,$$ where the center is at $(-g, -f)$ and the radius is $$r = \sqrt{g^2 + f^2 - c}.$$\n\n3. **Find the center of the given circle:** Rewrite the given circle's equation: $$x^2 + y^2 - 8x + 6y + 20 = 0.$$ Here, $2g = -8 \Rightarrow g = -4$ and $2f = 6 \Rightarrow f = 3$. So the center is at $$(-g, -f) = (4, -3).$$\n\n4. **Since the new circle is concentric, it has the same center:** The new circle's equation is $$x^2 + y^2 + 2gx + 2fy + c = 0,$$ with $g = -4$ and $f = 3$, so $$x^2 + y^2 - 8x + 6y + c = 0.$$\n\n5. **Find $c$ using the point $(5,4)$ on the new circle:** Substitute $x=5$, $y=4$ into the equation: $$5^2 + 4^2 - 8(5) + 6(4) + c = 0.$$\n\nCalculate step-by-step:\n$$25 + 16 - 40 + 24 + c = 0,$$\n$$25 + 16 = 41,$$\n$$41 - 40 = 1,$$\n$$1 + 24 = 25,$$\nSo, $$25 + c = 0 \Rightarrow c = -25.$$\n\n6. **Write the final equation:** $$x^2 + y^2 - 8x + 6y - 25 = 0.$$