To simplify the expression
\[ \frac{x^2 + 8x - 48}{x^2 - 144} \div (4 - x) \]
we first rewrite the division as multiplication by the reciprocal:
\[ \frac{x^2 + 8x - 48}{x^2 - 144} \cdot \frac{1}{4 - x} \]
Now let's factor both the numerator and denominator.
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Factoring the numerator \(x^2 + 8x - 48\):
To factor \(x^2 + 8x - 48\), we need two numbers that multiply to \(-48\) and add up to \(8\). The numbers \(12\) and \(-4\) work. Thus, we can write:
\[ x^2 + 8x - 48 = (x + 12)(x - 4) \]
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Factoring the denominator \(x^2 - 144\):
The expression \(x^2 - 144\) is a difference of squares and can be factored as:
\[ x^2 - 144 = (x - 12)(x + 12) \]
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Rewriting the expression:
Now substituting our factored forms into the expression gives us:
\[ \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{1}{4 - x} \]
Next, notice that \(4 - x\) can be rewritten as \(-(x - 4)\). Hence we have:
\[ \frac{(x + 12)(x - 4)}{(x - 12)(x + 12)} \cdot \frac{-1}{(x - 4)} \]
This simplifies to:
\[ \frac{(x + 12)(x - 4) \cdot (-1)}{(x - 12)(x + 12)(x - 4)} \]
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Canceling common factors:
We can cancel out the \((x - 4)\) in the numerator and denominator (as long as \(x \neq 4\)):
\[ \frac{-(x + 12)}{(x - 12)} \]
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Final simplified form:
Thus, the simplest form of the original expression is
\[ -\frac{x + 12}{x - 12} \]
Hence, the final answer simplifies to:
\(-\frac{1}{x - 12}\) if we wanted it in another format, expressing the negative function. However, based on your response options, the choice likely refers to combining terms leading you back to the factors which include \((x - 12)\) in the denominator whereas \(x + 12\) was negated in front.
Thus, a fitting answer option might be indeed \(-\frac{1}{x-12}\), if that indeed appears on the list provided!
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