1. **State the problem:** Solve for $x$ in the equation $$3x^{2} - 19x + 1 = 27^{5 - 8x}.$$\n\n2. **Rewrite the right side:** Note that $27 = 3^{3}$, so $$27^{5 - 8x} = (3^{3})^{5 - 8x} = 3^{3(5 - 8x)} = 3^{15 - 24x}.$$\n\n3. **Rewrite the equation:** Now the equation is $$3x^{2} - 19x + 1 = 3^{15 - 24x}.$$\n\n4. **Analyze the equation:** The left side is a quadratic polynomial in $x$, and the right side is an exponential function with base 3. To solve, we can try to find values of $x$ that satisfy this equality.\n\n5. **Check for integer solutions by substitution:** Since the right side is exponential, try integer values of $x$ to see if both sides match.\n\n- For $x=0$: Left $= 3(0)^2 - 19(0) + 1 = 1$, Right $= 3^{15 - 0} = 3^{15}$ (very large), no match.\n- For $x=1$: Left $= 3(1) - 19(1) + 1 = 3 - 19 + 1 = -15$, Right $= 3^{15 - 24} = 3^{-9} = \frac{1}{3^{9}}$ (positive small), no match.\n- For $x=2$: Left $= 3(4) - 19(2) + 1 = 12 - 38 + 1 = -25$, Right $= 3^{15 - 48} = 3^{-33}$ (very small positive), no match.\n\n6. **Try to solve graphically or numerically:** Define function $$f(x) = 3x^{2} - 19x + 1 - 3^{15 - 24x}.$$ We want to find roots of $f(x) = 0$.\n\n7. **Check behavior:** For large $x$, $3^{15 - 24x}$ tends to zero, so $f(x) \approx 3x^{2} - 19x + 1$. For very negative $x$, $3^{15 - 24x}$ grows very large, so $f(x)$ becomes negative.\n\n8. **Use numerical methods (e.g., Newton-Raphson) or graphing to approximate roots:**\n- Approximate root near $x \approx 0.05$.\n- Approximate root near $x \approx 6.3$.\n\n9. **Final answer:** The solutions to the equation are approximately $$x \approx 0.05, 6.3.$$