Find the Holes in the Graph f(x)=(5x+5)/(-x^2-2x-1)

$f (x) = \frac{5 x + 5}{- x^{2} - 2 x - 1}$

Factor $5$ out of $5 x + 5$ .

Factor $5$ out of $5 x$ .

$f (x) = \frac{5 (x) + 5}{- x^{2} - 2 x - 1}$

Factor $5$ out of $5$ .

$f (x) = \frac{5 (x) + 5 (1)}{- x^{2} - 2 x - 1}$

Factor $5$ out of $5 (x) + 5 (1)$ .

$f (x) = \frac{5 (x + 1)}{- x^{2} - 2 x - 1}$

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 $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = - 2$ .

Factor $- 2$ out of $- 2 x$ .

$f (x) = \frac{5 (x + 1)}{- x^{2} - 2 x - 1}$

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

$f (x) = \frac{5 (x + 1)}{- x^{2} + (- 1 - 1) x - 1}$

Apply the distributive property.

$f (x) = \frac{5 (x + 1)}{- x^{2} - 1 x - 1 x - 1}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = \frac{5 (x + 1)}{(- x^{2} - 1 x) - 1 x - 1}$

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

$f (x) = \frac{5 (x + 1)}{x (- x - 1) + 1 (- x - 1)}$

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

$f (x) = \frac{5 (x + 1)}{(- x - 1) (x + 1)}$

Cancel the common factor of $x + 1$ and $- x - 1$ .

Factor $- 1$ out of $x$ .

$f (x) = \frac{5 (- 1 (- x) + 1)}{(- x - 1) (x + 1)}$

Rewrite $1$ as $- 1 (- 1)$ .

$f (x) = \frac{5 (- 1 (- x) - 1 \cdot - 1)}{(- x - 1) (x + 1)}$

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

$f (x) = \frac{5 (- 1 (- x - 1))}{(- x - 1) (x + 1)}$

Cancel the common factor.

$f (x) = \frac{5 (- 1 (- x - 1))}{(- x - 1) (x + 1)}$

Rewrite the expression.

$f (x) = \frac{5 \cdot (- 1)}{x + 1}$

Simplify the numerator.

Multiply $5$ by $- 1$ .

$f (x) = \frac{5 (- 1)}{x + 1}$

Multiply $5$ by $- 1$ .

$f (x) = \frac{- 5}{x + 1}$

Move the negative in front of the fraction.

$f (x) = - \frac{5}{x + 1}$

Since no factors can be removed from the denominator, there are no holes in the graph.

No holes

Find the Holes in the Graph f(x)=(5x+5)/(-x^2-2x-1)

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