To solve the equation \(\frac{(x + 27)^2}{-6} = -3\), we first eliminate the fraction by multiplying both sides by \(-6\):
\[ (x + 27)^2 = -3 \times -6 \]
This simplifies to:
\[ (x + 27)^2 = 18 \]
Next, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{18} \]
Calculating \(\sqrt{18}\) gives:
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \approx 3 \times 1.414 = 4.242 \]
Now, we have two equations:
- \(x + 27 = 3\sqrt{2}\)
- \(x + 27 = -3\sqrt{2}\)
Solving for \(x\):
- For \(x + 27 = 3\sqrt{2}\):
\[ x = 3\sqrt{2} - 27 \approx 4.242 - 27 \approx -22.758 \]
- For \(x + 27 = -3\sqrt{2}\):
\[ x = -3\sqrt{2} - 27 \approx -4.242 - 27 \approx -31.242 \]
Final values:
- The smaller value (rounded to the nearest hundredth) is approximately \(-31.24\).
- The larger value (rounded to the nearest hundredth) is approximately \(-22.76\).
Therefore, the smaller of the two values first:
\[ \boxed{-31.24} \quad \text{and} \quad \boxed{-22.76} \]