We are tasked with finding the limit of the expression 

\[
\lim_{x \to -1} \frac{5 - \sqrt{3x + 28}}{x + 1}
\]

### Step 1: Direct Substitution
First, let's substitute \( x = -1 \) directly into the expression:

- Numerator: \( 5 - \sqrt{3(-1) + 28} = 5 - \sqrt{-3 + 28} = 5 - \sqrt{25} = 5 - 5 = 0 \)
- Denominator: \( x + 1 = -1 + 1 = 0 \)

So, direct substitution gives the indeterminate form \( \frac{0}{0} \). We need to resolve this indeterminate form by using an algebraic method like factoring, rationalizing, or applying L'Hopital's Rule.

### Step 2: Rationalizing the Numerator
Since we have a square root in the numerator, we can rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator:

\[
\frac{5 - \sqrt{3x + 28}}{x + 1} \cdot \frac{5 + \sqrt{3x + 28}}{5 + \sqrt{3x + 28}} = \frac{(5 - \sqrt{3x + 28})(5 + \sqrt{3x + 28})}{(x + 1)(5 + \sqrt{3x + 28})}
\]

Using the difference of squares formula, the numerator simplifies to:

\[
(5 - \sqrt{3x + 28})(5 + \sqrt{3x + 28}) = 5^2 - (\sqrt{3x + 28})^2 = 25 - (3x + 28) = 25 - 3x - 28 = -3x - 3
\]

Thus, the expression becomes:

\[
\frac{-3(x + 1)}{(x + 1)(5 + \sqrt{3x + 28})}
\]

### Step 3: Simplify the Expression
We can now cancel out the \( x + 1 \) terms:

\[
\frac{-3}{5 + \sqrt{3x + 28}}
\]

### Step 4: Apply the Limit
Now, we can substitute \( x = -1 \) into the simplified expression:

\[
\frac{-3}{5 + \sqrt{3(-1) + 28}} = \frac{-3}{5 + \sqrt{-3 + 28}} = \frac{-3}{5 + \sqrt{25}} = \frac{-3}{5 + 5} = \frac{-3}{10}
\]

Thus, the limit is:

\[
\boxed{-\frac{3}{10}}
\]

Let me know if you'd like more details on this solution!

### Related Questions:
1. What other methods can resolve indeterminate forms?
2. How does L'Hopital's Rule apply to limits?
3. What are the general steps for rationalizing a square root in a fraction?
4. How do you find limits involving radicals at infinity?
5. Can all indeterminate forms be resolved using rationalization?

### Tip:
Always check for simpler methods before jumping to more advanced techniques like L'Hopital’s Rule. Rationalizing is often quicker!

Explore how to solve the limit of (5 - sqrt(3x + 28)) / (x + 1) as x approaches -1, using rationalization and algebraic methods. This step-by-step guide covers indeterminate forms and the application of the difference of squares formula, perfect for high school and early college students.

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