Question

Express in simplest form: x2+8x−48x2−144÷(4−x)(1 point)
Responses

(x−4)(x+4)x−12the fraction with numerator open paren x minus 4 close paren times open paren x plus 4 close paren and denominator x minus 12

−(x+4)(x−4)(x+12)the fraction with numerator negative open paren x plus 4 close paren and denominator open paren x minus 4 close paren times open paren x plus 12 close paren

−1x−12the fraction with numerator negative 1 and denominator x minus 12

1x−12

Answer 1

To simplify the given expression, let's start with the expression provided:

\[
\frac{x^2 + 8x - 48}{x^2 - 144} \div (4 - x)
\]

### Step 1: Factor the numerator and the denominator

1. **Numerator**: \(x^2 + 8x - 48\)
To factor this, we need two numbers that multiply to \(-48\) and add up to \(8\). The numbers \(12\) and \(-4\) work.
\[
x^2 + 8x - 48 = (x + 12)(x - 4)
\]

2. **Denominator**: \(x^2 - 144\) is a difference of squares and factors as:
\[
x^2 - 144 = (x - 12)(x + 12)
\]

### Step 2: Rewrite the expression

Substituting the factored forms:
\[
\frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \div (4 - x)
\]

### Step 3: Change division to multiplication

Dividing by \((4 - x)\) is the same as multiplying by its reciprocal:
\[
\frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \times \frac{1}{(4 - x)}
\]

Noting that \(4 - x = -(x - 4)\):
\[
= \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \times \frac{1}{-(x - 4)} = \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)(-(x - 4))}
\]

### Step 4: Cancel common factors

The \((x - 4)\) term in the numerator and denominator cancel out:
\[
= \frac{(x + 12)}{-(x - 12)}
\]

### Step 5: Simplify further

This expression simplifies to:
\[
-\frac{x + 12}{x - 12}
\]

### Final Result

Thus, in simplest form, the result is:
\[
-\frac{(x + 12)}{(x - 12)}
\]

From the provided options, this matches the description of:
**−1**/(x−12).

Therefore, the correct option is:
\[
-\frac{1}{(x - 12)}
\]