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.