Let's break down the problem step-by-step.

The problem is:

$\frac{3k^2 - 2k - 1}{3k^2 + 10k + 7} \div \frac{9k^2 - 1}{3k^2 + 4k - 7}$

Step 1: Changing division into multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. So, we first rewrite the expression as:

$\frac{3k^2 - 2k - 1}{3k^2 + 10k + 7} \times \frac{3k^2 + 4k - 7}{9k^2 - 1}$

Step 2: Factorization of the quadratics

Now, let's factor each of the quadratics:

1. Factor $3k^2 - 2k - 1$: We need to find two numbers that multiply to $3 \times (-1) = -3$ and add to $-2$. These numbers are $-3$ and $1$.

   Therefore: $3k^2 - 2k - 1 = (3k + 1)(k - 1)$

2. Factor $3k^2 + 10k + 7$: We need two numbers that multiply to $3 \times 7 = 21$ and add to $10$. These numbers are $3$ and $7$.

   Therefore: $3k^2 + 10k + 7 = (3k + 7)(k + 1)$

3. Factor $9k^2 - 1$ (difference of squares): $9k^2 - 1 = (3k - 1)(3k + 1)$

4. Factor $3k^2 + 4k - 7$: We need two numbers that multiply to $3 \times (-7) = -21$ and add to $4$. These numbers are $7$ and $-3$.

   Therefore: $3k^2 + 4k - 7 = (3k - 7)(k + 1)$

Step 3: Substitute the factors into the expression

Now, we can substitute the factors we found into the expression:

$\frac{(3k + 1)(k - 1)}{(3k + 7)(k + 1)} \times \frac{(3k - 7)(k + 1)}{(3k - 1)(3k + 1)}$

Step 4: Simplify the expression

We can cancel out the common factors in the numerator and denominator:

• $(3k + 1)$ appears in both the numerator and denominator, so it cancels.
• $(k + 1)$ appears in both the numerator and denominator, so it cancels.

The remaining expression is:

$\frac{(k - 1)(3k - 7)}{(3k + 7)(3k - 1)}$

Step 5: Final Answer

The simplified expression in lowest terms is:

$\frac{(k - 1)(3k - 7)}{(3k + 7)(3k - 1)}$

Would you like more explanation or further steps on how to work with rational expressions?

Here are some related questions you can explore:

1. How do you factor trinomials when the leading coefficient is not 1?
2. What does it mean for rational expressions to be in lowest terms?
3. How do you solve rational equations involving polynomials?
4. How can you perform polynomial long division to simplify expressions?
5. What are the common mistakes when factoring quadratics?

Tip: Always check for common factors across the numerator and denominator in rational expressions before simplifying!