Simplify (x^3+15x^2+54x)/(2x^4+6x^3-108x^2)*(6x^2-42x+36)/(x^2+5x-6)

$\frac{x^{3} + 15 x^{2} + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Simplify the numerator.

Factor $x$ out of $x^{3} + 15 x^{2} + 54 x$ .

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

$\frac{x \cdot x^{2} + 15 x^{2} + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x$ out of $15 x^{2}$ .

$\frac{x \cdot x^{2} + x (15 x) + 54 x}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x$ out of $54 x$ .

$\frac{x \cdot x^{2} + x (15 x) + x \cdot 54}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x$ out of $x \cdot x^{2} + x (15 x)$ .

$\frac{x (x^{2} + 15 x) + x \cdot 54}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x$ out of $x (x^{2} + 15 x) + x \cdot 54$ .

$\frac{x (x^{2} + 15 x + 54)}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x^{2} + 15 x + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Write the factored form using these integers.

$\frac{x ((x + 6) (x + 9))}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

$\frac{x (x + 6) (x + 9)}{2 x^{4} + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Simplify the denominator.

Factor $2 x^{2}$ out of $2 x^{4} + 6 x^{3} - 108 x^{2}$ .

Factor $2 x^{2}$ out of $2 x^{4}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 6 x^{3} - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $2 x^{2}$ out of $6 x^{3}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 2 x^{2} (3 x) - 108 x^{2}} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $2 x^{2}$ out of $- 108 x^{2}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2}) + 2 x^{2} (3 x) + 2 x^{2} (- 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

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

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2} + 3 x) + 2 x^{2} (- 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $2 x^{2}$ out of $2 x^{2} (x^{2} + 3 x) + 2 x^{2} (- 54)$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x^{2} + 3 x - 54)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $x^{2} + 3 x - 54$ 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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Write the factored form using these integers.

$\frac{x (x + 6) (x + 9)}{2 x^{2} ((x - 6) (x + 9))} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 x^{2} - 42 x + 36}{x^{2} + 5 x - 6}$

Simplify the numerator.

Factor $6$ out of $6 x^{2} - 42 x + 36$ .

Factor $6$ out of $6 x^{2}$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2}) - 42 x + 36}{x^{2} + 5 x - 6}$

Factor $6$ out of $- 42 x$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2}) + 6 (- 7 x) + 36}{x^{2} + 5 x - 6}$

Factor $6$ out of $36$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 x^{2} + 6 (- 7 x) + 6 \cdot 6}{x^{2} + 5 x - 6}$

Factor $6$ out of $6 x^{2} + 6 (- 7 x)$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2} - 7 x) + 6 \cdot 6}{x^{2} + 5 x - 6}$

Factor $6$ out of $6 (x^{2} - 7 x) + 6 \cdot 6$ .

$\frac{x (x + 6) (x + 9)}{2 x^{2} (x - 6) (x + 9)} \cdot \frac{6 (x^{2} - 7 x + 6)}{x^{2} + 5 x - 6}$

Factor $x^{2} - 7 x + 6$ 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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Write the factored form using these integers.

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

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

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Write the factored form using these integers.

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

Simplify terms.

Combine.

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

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $x (x + 6) (x + 9) (6 (x - 6) (x - 1))$ .

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

Cancel the common factors.

Factor $x$ out of $2 x^{2} (x - 6) (x + 9) ((x - 1) (x + 6))$ .

$\frac{x (((x + 6) (x + 9)) (6 (x - 6) (x - 1)))}{x (2 x (x - 6) (x + 9) ((x - 1) (x + 6)))}$

Cancel the common factor.

$\frac{x (((x + 6) (x + 9)) (6 (x - 6) (x - 1)))}{x (2 x (x - 6) (x + 9) ((x - 1) (x + 6)))}$

Rewrite the expression.

$\frac{((x + 6) (x + 9)) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) ((x - 1) (x + 6))}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x + 6) (x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) ((x - 1) (x + 6))}$

Rewrite the expression.

$\frac{(x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) (x - 1)}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{(x + 9) (6 (x - 6) (x - 1))}{2 x (x - 6) (x + 9) (x - 1)}$

Rewrite the expression.

$\frac{6 (x - 6) (x - 1)}{2 x (x - 6) (x - 1)}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 (x - 6) (x - 1)$ .

$\frac{2 (3 (x - 6) (x - 1))}{2 x (x - 6) (x - 1)}$

Cancel the common factors.

Factor $2$ out of $2 x (x - 6) (x - 1)$ .

$\frac{2 (3 (x - 6) (x - 1))}{2 ((x (x - 6)) (x - 1))}$

Cancel the common factor.

$\frac{2 (3 (x - 6) (x - 1))}{2 ((x (x - 6)) (x - 1))}$

Rewrite the expression.

$\frac{3 (x - 6) (x - 1)}{(x (x - 6)) (x - 1)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{3 (x - 6) (x - 1)}{x (x - 6) (x - 1)}$

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3}{x}$

Simplify (x^3+15x^2+54x)/(2x^4+6x^3-108x^2)*(6x^2-42x+36)/(x^2+5x-6)

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