Simplify ((9x^2-1)/(12x^2-12x))÷((9x^2+12x+3)/(3x^2-6x+3))

$\frac{9 x^{2} - 1}{12 x^{2} - 12 x} \div \frac{9 x^{2} + 12 x + 3}{3 x^{2} - 6 x + 3}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x^{2} - 1}{12 x^{2} - 12 x} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Simplify the numerator.

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{{(3 x)}^{2} - 1}{12 x^{2} - 12 x} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Rewrite $1$ as $1^{2}$ .

$\frac{{(3 x)}^{2} - 1^{2}}{12 x^{2} - 12 x} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 1$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x^{2} - 12 x} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Simplify terms.

Factor $12 x$ out of $12 x^{2} - 12 x$ .

Factor $12 x$ out of $12 x^{2}$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x) - 12 x} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Factor $12 x$ out of $- 12 x$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x) + 12 x (- 1)} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Factor $12 x$ out of $12 x (x) + 12 x (- 1)$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 x^{2} - 6 x + 3}{9 x^{2} + 12 x + 3}$

Cancel the common factor of $3 x^{2} - 6 x + 3$ and $9 x^{2} + 12 x + 3$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2}) - 6 x + 3}{9 x^{2} + 12 x + 3}$

Factor $3$ out of $- 6 x$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2}) + 3 (- 2 x) + 3}{9 x^{2} + 12 x + 3}$

Factor $3$ out of $3 (x^{2}) + 3 (- 2 x)$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x) + 3}{9 x^{2} + 12 x + 3}$

Factor $3$ out of $3$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x) + 3 \cdot 1}{9 x^{2} + 12 x + 3}$

Factor $3$ out of $3 (x^{2} - 2 x) + 3 (1)$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{9 x^{2} + 12 x + 3}$

Cancel the common factors.

Factor $3$ out of $9 x^{2}$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2}) + 12 x + 3}$

Factor $3$ out of $12 x$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2}) + 3 (4 x) + 3}$

Factor $3$ out of $3 (3 x^{2}) + 3 (4 x)$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2} + 4 x) + 3}$

Factor $3$ out of $3$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2} + 4 x) + 3 (1)}$

Factor $3$ out of $3 (3 x^{2} + 4 x) + 3 (1)$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2} + 4 x + 1)}$

Cancel the common factor.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{3 (x^{2} - 2 x + 1)}{3 (3 x^{2} + 4 x + 1)}$

Rewrite the expression.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{x^{2} - 2 x + 1}{3 x^{2} + 4 x + 1}$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{x^{2} - 2 x + 1^{2}}{3 x^{2} + 4 x + 1}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}{3 x^{2} + 4 x + 1}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{3 x^{2} + 4 x + 1}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = 4$ .

Factor $4$ out of $4 x$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{3 x^{2} + 4 (x) + 1}$

Rewrite $4$ as $1$ plus $3$

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{3 x^{2} + (1 + 3) x + 1}$

Apply the distributive property.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{3 x^{2} + 1 x + 3 x + 1}$

Multiply $x$ by $1$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{3 x^{2} + x + 3 x + 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{(3 x^{2} + x) + 3 x + 1}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{x (3 x + 1) + 1 (3 x + 1)}$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{(3 x + 1) (3 x - 1)}{12 x (x - 1)} \cdot \frac{{(x - 1)}^{2}}{(3 x + 1) (x + 1)}$

Combine.

$\frac{(3 x + 1) (3 x - 1) {(x - 1)}^{2}}{12 x (x - 1) ((3 x + 1) (x + 1))}$

Cancel the common factor of $3 x + 1$ .

Cancel the common factor.

$\frac{(3 x + 1) (3 x - 1) {(x - 1)}^{2}}{12 x (x - 1) ((3 x + 1) (x + 1))}$

Rewrite the expression.

$\frac{(3 x - 1) {(x - 1)}^{2}}{12 x (x - 1) (x + 1)}$

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

Factor $x - 1$ out of $(3 x - 1) {(x - 1)}^{2}$ .

$\frac{(x - 1) ((3 x - 1) (x - 1))}{12 x (x - 1) (x + 1)}$

Cancel the common factors.

Factor $x - 1$ out of $12 x (x - 1) (x + 1)$ .

$\frac{(x - 1) ((3 x - 1) (x - 1))}{(x - 1) (12 x (x + 1))}$

Cancel the common factor.

$\frac{(x - 1) ((3 x - 1) (x - 1))}{(x - 1) (12 x (x + 1))}$

Rewrite the expression.

$\frac{(3 x - 1) (x - 1)}{12 x (x + 1)}$

Simplify ((9x^2-1)/(12x^2-12x))÷((9x^2+12x+3)/(3x^2-6x+3))

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