1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{8x^2 + 6x - 1}{\sqrt{x^2 + 5x + 2}}.$$\n\n2. **Substitute $x=2$ directly:**\nCalculate numerator: $8(2)^2 + 6(2) - 1 = 8 \times 4 + 12 - 1 = 32 + 12 - 1 = 43.$\nCalculate denominator: $\sqrt{(2)^2 + 5(2) + 2} = \sqrt{4 + 10 + 2} = \sqrt{16} = 4.$\n\n3. **Evaluate the limit:**\nSince the denominator is not zero and the expression is defined at $x=2$, the limit is simply the value of the function at $x=2$.\n$$\lim_{x \to 2} \frac{8x^2 + 6x - 1}{\sqrt{x^2 + 5x + 2}} = \frac{43}{4}.$$\n\n4. **Final answer:**\n$$\boxed{\frac{43}{4}}.$$