Simplify (x^2-9)/(6x^2+x-1)(2x^2+5x+2)/(x^2+4x+3)(x^2-x-2)/(x^2-3x)

$\frac{x^{2} - 9}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{x^{2} - x - 2}{x^{2} - 3 x}$

Factor $x^{2} - x - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$\frac{x^{2} - 9}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x^{2} - 3 x}$

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

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

$\frac{x^{2} - 9}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x \cdot x - 3 x}$

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

$\frac{x^{2} - 9}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x \cdot x + x \cdot - 3}$

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$\frac{x^{2} - 9}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

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

$\frac{(x + 3) (x - 3)}{6 x^{2} + x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

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 $a \cdot c = 6 \cdot - 1 = - 6$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{(x + 3) (x - 3)}{6 x^{2} + 1 x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

Rewrite $1$ as $- 2$ plus $3$

$\frac{(x + 3) (x - 3)}{6 x^{2} + (- 2 + 3) x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

Apply the distributive property.

$\frac{(x + 3) (x - 3)}{6 x^{2} - 2 x + 3 x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x + 3) (x - 3)}{(6 x^{2} - 2 x) + 3 x - 1} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

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

$\frac{(x + 3) (x - 3)}{2 x (3 x - 1) + 1 (3 x - 1)} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

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

$\frac{(x + 3) (x - 3)}{(3 x - 1) (2 x + 1)} \cdot \frac{2 x^{2} + 5 x + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{(x + 3) (x - 3)}{(3 x - 1) (2 x + 1)} \cdot \frac{2 x^{2} + 5 (x) + 2}{x^{2} + 4 x + 3} \cdot \frac{(x - 2) (x + 1)}{x (x - 3)}$

Rewrite $5$ as $1$ plus $4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Factor $x^{2} + 4 x + 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Simplify terms.

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $(x + 1) (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x + 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x - 3}{3 x - 1}$ by $\frac{x + 2}{x + 1}$ .

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

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $x (x - 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $(3 x - 1) (x + 1)$ .

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

Factor $x + 1$ out of $(x - 2) (x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 2}{3 x - 1} \cdot \frac{x - 2}{x}$

Multiply $\frac{x + 2}{3 x - 1}$ by $\frac{x - 2}{x}$ .

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

Reorder factors in $\frac{(x + 2) (x - 2)}{(3 x - 1) x}$ .

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

Simplify (x^2-9)/(6x^2+x-1)*(2x^2+5x+2)/(x^2+4x+3)*(x^2-x-2)/(x^2-3x)

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Simplify (x^2-9)/(6x^2+x-1)(2x^2+5x+2)/(x^2+4x+3)(x^2-x-2)/(x^2-3x)