Simplify (x^2+3x-10)/(6(x+5)^2)*(x^4+3x^3-10x^2)/(x(x-2)^2)

$\frac{x^{2} + 3 x - 10}{6 {(x + 5)}^{2}} \cdot \frac{x^{4} + 3 x^{3} - 10 x^{2}}{x {(x - 2)}^{2}}$

Step 1

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{4} + 3 x^{3} - 10 x^{2}}{x {(x - 2)}^{2}}$

Step 2

Simplify the numerator.

Factor $x^{2}$ out of $x^{4} + 3 x^{3} - 10 x^{2}$ .

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

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{2} x^{2} + 3 x^{3} - 10 x^{2}}{x {(x - 2)}^{2}}$

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

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{2} x^{2} + x^{2} (3 x) - 10 x^{2}}{x {(x - 2)}^{2}}$

Factor $x^{2}$ out of $- 10 x^{2}$ .

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot - 10}{x {(x - 2)}^{2}}$

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

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{2} (x^{2} + 3 x) + x^{2} \cdot - 10}{x {(x - 2)}^{2}}$

Factor $x^{2}$ out of $x^{2} (x^{2} + 3 x) + x^{2} \cdot - 10$ .

$\frac{(x - 2) (x + 5)}{6 {(x + 5)}^{2}} \cdot \frac{x^{2} (x^{2} + 3 x - 10)}{x {(x - 2)}^{2}}$

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

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

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

Step 3

Simplify terms.

Combine.

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

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

Factor $x - 2$ out of $(x - 2) (x + 5) (x^{2} (x - 2) (x + 5))$ .

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

Cancel the common factors.

Factor $x - 2$ out of $6 {(x + 5)}^{2} (x {(x - 2)}^{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Cancel the common factors.

Factor $x + 5$ out of $6 {(x + 5)}^{2} (x (x - 2))$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

Factor $x$ out of $x^{2} (x - 2) (x + 5)$ .

$\frac{x ((x (x - 2)) (x + 5))}{6 (x + 5) (x (x - 2))}$

Cancel the common factors.

Factor $x$ out of $6 (x + 5) (x (x - 2))$ .

$\frac{x ((x (x - 2)) (x + 5))}{x (6 (x + 5) (x - 2))}$

Cancel the common factor.

$\frac{x ((x (x - 2)) (x + 5))}{x (6 (x + 5) (x - 2))}$

Rewrite the expression.

$\frac{(x (x - 2)) (x + 5)}{6 (x + 5) (x - 2)}$

Step 6

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{x (x - 2) (x + 5)}{6 (x + 5) (x - 2)}$

Rewrite the expression.

$\frac{(x) (x + 5)}{6 (x + 5)}$

Step 7

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{x (x + 5)}{6 (x + 5)}$

Rewrite the expression.

$\frac{x}{6}$

Simplify (x^2+3x-10)/(6(x+5)^2)*(x^4+3x^3-10x^2)/(x(x-2)^2)

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