Question

What is the least common denominator of the equation 5/x+5−1/x^2+2x−15=4/x^2+2x−15
?(1 point)
Responses

(x+5)(x^2+2x−15)
left parenthesis x plus 5 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis

(x+5)(x−3)
left parenthesis x plus 5 right parenthesis left parenthesis x minus 3 right parenthesis

(x+5)(x^2+2x−15)(x^2+2x−15)
left parenthesis x plus 5 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis

(x+5)
left parenthesis x plus 5 right parenthesis
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Answer 1

To find the least common denominator (LCD) of the equation

\[
\frac{5}{x+5} - \frac{1}{x^2 + 2x - 15} = \frac{4}{x^2 + 2x - 15}
\]

we need to determine the denominators involved:

1. \(x + 5\)
2. \(x^2 + 2x - 15\)

Next, we factor \(x^2 + 2x - 15\):

\[
x^2 + 2x - 15 = (x + 5)(x - 3)
\]

Now we can identify the denominators as follows:

- The denominator \(x + 5\) appears as-is.
- The denominator \(x^2 + 2x - 15\) factors to \((x + 5)(x - 3)\).

To find the least common denominator, we take the highest power of each factor present in the denominators:

- From \(x + 5\): we take it once.
- From \((x + 5)(x - 3)\): we take both \(x + 5\) and \(x - 3\).

Therefore, the least common denominator (LCD) is:

\[
(x + 5)(x - 3)
\]

Thus, the correct response is:

\[
(x + 5)(x - 3)
\]