Find the Excluded Values ((x^2-16)/(x^2+5x+6))÷((x^2+5x+4)/(x^2-2x-8))

$\frac{x^{2} - 16}{x^{2} + 5 x + 6} \div \frac{x^{2} + 5 x + 4}{x^{2} - 2 x - 8}$

Step 1

Set the denominator in $\frac{x^{2} - 16}{x^{2} + 5 x + 6}$ equal to $0$ to find where the expression is undefined.

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

Step 2

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 3

Set the denominator in $\frac{x^{2} + 5 x + 4}{x^{2} - 2 x - 8}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 2 x - 8 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 2 x - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

$(x - 4) (x + 2) = 0$

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

$x - 4 = 0$

$x + 2 = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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.

$x = - 2$

The final solution is all the values that make $(x - 4) (x + 2) = 0$ true.

$x = 4, - 2$

Step 5

Set the denominator in $\frac{\frac{x^{2} - 16}{x^{2} + 5 x + 6}}{\frac{x^{2} + 5 x + 4}{x^{2} - 2 x - 8}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} + 5 x + 4}{x^{2} - 2 x - 8} = 0$

Step 6

Solve for $x$ .

Set the numerator equal to zero.

$x^{2} + 5 x + 4 = 0$

Solve the equation for $x$ .

Factor $x^{2} + 5 x + 4$ 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 $4$ and whose sum is $5$ .

$1, 4$

Write the factored form using these integers.

$(x + 1) (x + 4) = 0$

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

$x + 1 = 0$

$x + 4 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x + 1) (x + 4) = 0$ true.

$x = - 1, - 4$

Step 7

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 $0$ .

$x = - 4, x = - 3, x = - 2, x = - 1, x = 4$

Step 8

Find the Excluded Values ((x^2-16)/(x^2+5x+6))÷((x^2+5x+4)/(x^2-2x-8))

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