Find the Excluded Values (x^2-x-6)/(4x^3)*(2x^2+2x)/(x^2+5x+6)

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

Step 1

Set the denominator in $\frac{x^{2} - x - 6}{4 x^{3}}$ equal to $0$ to find where the expression is undefined.

$4 x^{3} = 0$

Step 2

Solve for $x$ .

Divide each term in $4 x^{3} = 0$ by $4$ and simplify.

Divide each term in $4 x^{3} = 0$ by $4$ .

$\frac{4 x^{3}}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{3}}{4} = \frac{0}{4}$

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x^{3} = 0$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 3

Set the denominator in $\frac{2 x^{2} + 2 x}{x^{2} + 5 x + 6}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 5 x + 6 = 0$

Step 4

Solve for $x$ .

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 $5$ .

$2, 3$

Write the factored form using these integers.

$(x + 2) (x + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 3 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x + 2) (x + 3) = 0$ true.

$x = - 2, - 3$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 3, x = - 2, x = 0$

Step 6

Find the Excluded Values (x^2-x-6)/(4x^3)*(2x^2+2x)/(x^2+5x+6)

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