To simplify the rational expression
\[ \frac{hj - hp - 2j + 2p}{hj - hp + 2j - 2p} \]
we can start by rearranging the terms in both the numerator and the denominator.
Numerator: \[ hj - hp - 2j + 2p \quad \text{can be rearranged as} \quad hj - hp + 2p - 2j \]
Denominator: \[ hj - hp + 2j - 2p \quad \text{can also be rearranged as} \quad hj - hp + 2j - 2p \]
Now let's factor both the numerator and the denominator.
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Factoring the numerator: \[ hj - hp + 2p - 2j = h(j - p) + 2(p - j) \] Notice that \(2(p - j) = -2(j - p)\), so we can group terms: \[ = h(j - p) - 2(j - p) = (j - p)(h - 2) \]
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Factoring the denominator: \[ hj - hp + 2j - 2p = h(j - p) + 2(j - p) = (j - p)(h + 2) \]
Now substituting back into the original rational expression:
\[ \frac{(j - p)(h - 2)}{(j - p)(h + 2)} \]
Assuming \(j \neq p\) (to avoid division by zero), we can cancel out the common factor \((j - p)\):
\[ \frac{h - 2}{h + 2} \]
Thus, the rational expression in lowest terms is:
\[ \frac{h - 2}{h + 2} \]
This is the final simplified form.