`(a^(4)-a^(2)) div (a^(3) + a^(2))` = ______

Options

A. `(a^(2) +1)`
B. `a-1`
C. `a+1`
D. `a^(2)-1`

Answer

To solve the expression \((a^4 - a^2) \div (a^3 + a^2)\), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression in fraction form:
\[ \frac{a^4 - a^2}{a^3 + a^2} \]

### Step 2: Factor the numerator
Next, we factor the numerator \(a^4 - a^2\). We notice that \(a^2\) is a common factor:
\[ a^4 - a^2 = a^2(a^2 - 1) \]

### Step 3: Factor the denominator
Now, we factor the denominator \(a^3 + a^2\). Here, \(a^2\) is also a common factor:
\[ a^3 + a^2 = a^2(a + 1) \]

### Step 4: Substitute the factored forms back into the expression
Now we substitute the factored forms back into the expression:
\[ \frac{a^2(a^2 - 1)}{a^2(a + 1)} \]

### Step 5: Cancel common factors
We can now cancel the common factor \(a^2\) from the numerator and the denominator:
\[ \frac{a^2 - 1}{a + 1} \]

### Step 6: Factor \(a^2 - 1\)
The expression \(a^2 - 1\) can be factored further using the difference of squares:
\[ a^2 - 1 = (a - 1)(a + 1) \]

### Step 7: Substitute back into the expression
Substituting this back into our expression gives:
\[ \frac{(a - 1)(a + 1)}{a + 1} \]

### Step 8: Cancel the common factor again
We can cancel the common factor \(a + 1\):
\[ a - 1 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{a - 1} \]

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