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To solve this problem, let's divide the given rational expressions step-by-step and express the answer in its simplest form.

### Step 1: Understand the problem
The problem is:

\[
\frac{5k^2 - 4k - 1}{5k^2 + 16k + 11} \div \frac{25k^2 - 1}{5k^2 + 6k - 11}
\]

### Step 2: Change the division into multiplication
When dividing by a fraction, we multiply by its reciprocal:

\[
\frac{5k^2 - 4k - 1}{5k^2 + 16k + 11} \times \frac{5k^2 + 6k - 11}{25k^2 - 1}
\]

### Step 3: Factorize each expression
Let's factor each polynomial:

1. **Factor \(5k^2 - 4k - 1\):**
   - This can be written as \((5k + 1)(k - 1)\).

2. **Factor \(5k^2 + 16k + 11\):**
   - This can be written as \((5k + 11)(k + 1)\).

3. **Factor \(25k^2 - 1\):**
   - This is a difference of squares: \((5k - 1)(5k + 1)\).

4. **Factor \(5k^2 + 6k - 11\):**
   - This polynomial does not factorize further neatly with integer coefficients, so we'll keep it as is.

### Step 4: Substitute the factored expressions
Substituting the factored expressions, we have:

\[
\frac{(5k + 1)(k - 1)}{(5k + 11)(k + 1)} \times \frac{5k^2 + 6k - 11}{(5k - 1)(5k + 1)}
\]

### Step 5: Simplify the expression
We cancel common factors:

- The term \( (5k + 1) \) appears in both the numerator and the denominator, so they cancel out.

The simplified expression is:

\[
\frac{(k - 1)(5k^2 + 6k - 11)}{(5k + 11)(k + 1)(5k - 1)}
\]

### Final Answer
\[
\frac{(k - 1)(5k^2 + 6k - 11)}{(5k + 11)(k + 1)(5k - 1)}
\]

This is the answer in its lowest terms.

Would you like further details or have any questions?

### Related Questions
1. How do you multiply rational expressions?
2. How can you identify the difference of squares in a polynomial?
3. What steps can you take to factor polynomials with higher degrees?
4. How do you handle rational expressions with unfactorable polynomials?
5. What are the general rules for simplifying complex fractions?

### Tip
Always remember that when dividing by a fraction, multiplying by its reciprocal is the key to solving the expression accurately.

Divide. Express your answer in lowest terms: \( \frac{5k^2 - 4k - 1}{5k^2 + 16k + 11} \div \frac{25k^2 - 1}{5k^2 + 6k - 11} \).

Learn how to divide and simplify rational expressions involving polynomials. This problem covers key concepts such as factoring quadratics and using the difference of squares formula. Suitable for students in Grades 10-12.

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