Certainly! Let's carefully work through the problem step-by-step. We want to simplify the given quotient:
[tex]$
\frac{3 x^2-27 x}{2 x^2+13 x-7} \div \frac{3 x}{4 x^2-1}
$[/tex]
### Step 1: Understand the Division of Fractions
Dividing by a fraction is the same as multiplying by its reciprocal. This transforms our problem into:
[tex]$
\frac{3 x^2-27 x}{2 x^2+13 x-7} \times \frac{4 x^2-1}{3 x}
$[/tex]
### Step 2: Simplify the Numerators and Denominators
Let's factor the expressions wherever possible to simplify the multiplication.
#### Factoring the Numerators:
1. Numerator of the first fraction: [tex]\( 3 x^2 - 27 x \)[/tex]
[tex]\[
3 x^2 - 27 x = 3 x (x - 9)
\][/tex]
2. Numerator of the second fraction: [tex]\( 4 x^2 - 1 \)[/tex]
[tex]\[
4 x^2 - 1 = (2 x + 1)(2 x - 1)
\][/tex]
This is a difference of squares.
#### Factoring the Denominators:
1. Denominator of the first fraction: [tex]\( 2 x^2 + 13 x - 7 \)[/tex]
To factor [tex]\( 2 x^2 + 13 x - 7 \)[/tex], we need to find factors of [tex]\(-14\)[/tex] (product of [tex]\(2\)[/tex] and [tex]\(-7\)[/tex]) that add up to [tex]\(13\)[/tex].
[tex]\[
2 x^2 + 13 x - 7 = (2 x - 1)(x + 7)
\][/tex]
2. Denominator of the second fraction: [tex]\( 3 x \)[/tex]
This is already in its simplest form.
### Step 3: Rewrite the Problem with Factored Forms
Rewriting the fractions with the factored forms gives:
[tex]$
\frac{3 x (x - 9)}{(2 x - 1)(x + 7)} \times \frac{(2 x + 1)(2 x - 1)}{3 x}
$[/tex]
### Step 4: Multiply the Fractions
Multiply the numerators together and the denominators together:
[tex]$
\frac{3 x (x - 9) \cdot (2 x + 1)(2 x - 1)}{(2 x - 1)(x + 7) \cdot 3 x}
$[/tex]
### Step 5: Cancel Common Factors
Now, we look to cancel any common factors from the numerator and the denominator:
- [tex]\(3 x\)[/tex] in the numerator and denominator.
- [tex]\((2 x - 1)\)[/tex] in the numerator and denominator.
After canceling the common factors, we are left with:
[tex]$
\frac{(x - 9)(2 x + 1)}{x + 7}
$[/tex]
### Step 6: Expand or Simplify
Our expression is now simplified to:
[tex]$
\frac{(x - 9)(2 x + 1)}{x + 7}
$[/tex]
We can leave it in factored form or expand the numerator:
Expanding the numerator:
[tex]$
(x - 9)(2 x + 1) = x \cdot 2x + x \cdot 1 - 9 \cdot 2x - 9 \cdot 1 = 2 x^2 + x - 18 x - 9 = 2 x^2 - 17 x - 9
$[/tex]
So the simplified form is:
[tex]$
\frac{2 x^2 - 17 x - 9}{x + 7}
$[/tex]
Thus, the final simplified quotient is:
[tex]$
\boxed{\frac{2 x^2 - 17 x - 9}{x + 7}}
$[/tex]
