1. **State the problem:** Find the limit $$\lim_{x \to -3} \frac{x+3}{4 - \sqrt{2x + 22}}.$$\n\n2. **Check direct substitution:** Substitute $x = -3$:\n$$\frac{-3+3}{4 - \sqrt{2(-3) + 22}} = \frac{0}{4 - \sqrt{-6 + 22}} = \frac{0}{4 - \sqrt{16}} = \frac{0}{4 - 4} = \frac{0}{0}.$$\nThis is an indeterminate form, so we need to simplify.\n\n3. **Use conjugate to simplify:** Multiply numerator and denominator by the conjugate of the denominator:\n$$\frac{x+3}{4 - \sqrt{2x + 22}} \times \frac{4 + \sqrt{2x + 22}}{4 + \sqrt{2x + 22}} = \frac{(x+3)(4 + \sqrt{2x + 22})}{(4)^2 - (\sqrt{2x + 22})^2}.$$\n\n4. **Simplify denominator:**\n$$16 - (2x + 22) = 16 - 2x - 22 = -2x - 6.$$\n\n5. **Rewrite the expression:**\n$$\frac{(x+3)(4 + \sqrt{2x + 22})}{-2x - 6}.$$\n\n6. **Factor denominator:**\n$$-2x - 6 = -2(x + 3).$$\n\n7. **Cancel common factor $(x+3)$:**\n$$\frac{\cancel{(x+3)}(4 + \sqrt{2x + 22})}{-2\cancel{(x+3)}} = \frac{4 + \sqrt{2x + 22}}{-2}.$$\n\n8. **Evaluate the limit now:** Substitute $x = -3$:\n$$\frac{4 + \sqrt{2(-3) + 22}}{-2} = \frac{4 + \sqrt{-6 + 22}}{-2} = \frac{4 + \sqrt{16}}{-2} = \frac{4 + 4}{-2} = \frac{8}{-2} = -4.$$\n\n**Final answer:** The limit is $-4$.