Final answer:
The missing term to replace the question mark in the equation (g/(g²-h²)) - (?/(g-h)²) = (g²-2gh-h²)/((g-h)²(g+h)) is gh+h². By finding a common denominator and matching the numerators, we see that this term allows the left side of the equation to be equivalent to the right side.
Step-by-step explanation:
To solve for the missing term in the equation (g/(g²-h²)) - (?/(g-h)²) = (g²-2gh-h²)/((g-h)²(g+h)), we begin by finding a common denominator for the left side of the equation.
Since the denominator of the first fraction is (g²-h²), which is equivalent to (g+h)(g-h), and the denominator of the second term we are looking for is (g-h)², we note that (g+h)(g-h) is the factored form of the common denominator we have on the right side of the equation.
To create a single fraction on the left, the missing term in the numerator for the second fraction must allow the two terms to combine into the numerator on the right side, which is (g²-2gh-h²).
By inspection, the missing term must be complementary to what is already present in the numerator of the first fraction after having expanded the common denominator.
To do this, multiply the numerator and denominator of the first fraction by (g-h) to match the common denominator.
Doing so, for the first fraction we have: g(g-h)/(g+h)(g-h)² = (g²-gh)/(g-h)²(g+h).
Now, by comparison to the final numerator after combining the fractions, (g²-gh) - ? = g²-2gh-h² must hold true.
Therefore, the missing term must be gh+h², so that when subtracted from g²-gh, we obtain the correct numerator on the right side of the equation.
Thus, the term we need to replace the question mark is gh+h².