The simplest form of the expression ` (p^(2) - p)/( 2 p^(3) + 6 p^(2)) + (p^(2) - 1)/( p^(2) + 3 p) + (p^(2))/( p + 1)` is

Options

A. `2p^(2)`
B. `(1)/(2 p^(2))`
C. p + 3
D. `(1)/(p + 3)`

Answer

To simplify the expression \(\frac{p^2 - p}{2p^3 + 6p^2} + \frac{p^2 - 1}{p^2 + 3p} + \frac{p^2}{p + 1}\), we will follow these steps:

### Step 1: Factor the Numerators and Denominators
1. **First Fraction**:
- Numerator: \(p^2 - p = p(p - 1)\)
- Denominator: \(2p^3 + 6p^2 = 2p^2(p + 3)\)

Thus, the first fraction becomes:
\[ \frac{p(p - 1)}{2p^2(p + 3)} \]

2. **Second Fraction**:
- Numerator: \(p^2 - 1 = (p - 1)(p + 1)\)
- Denominator: \(p^2 + 3p = p(p + 3)\)

Thus, the second fraction becomes:
\[ \frac{(p - 1)(p + 1)}{p(p + 3)} \]

3. **Third Fraction**:
- Numerator: \(p^2\)
- Denominator: \(p + 1\)

Thus, the third fraction remains:
\[ \frac{p^2}{p + 1} \]

### Step 2: Rewrite the Expression
Now we can rewrite the entire expression:
\[ \frac{p(p - 1)}{2p^2(p + 3)} + \frac{(p - 1)(p + 1)}{p(p + 3)} + \frac{p^2}{p + 1} \]

### Step 3: Find a Common Denominator
The common denominator for the fractions is \(2p^2(p + 3)(p + 1)\).

### Step 4: Rewrite Each Fraction with the Common Denominator
1. **First Fraction**:
\[ \frac{p(p - 1) \cdot (p + 1)}{2p^2(p + 3)(p + 1)} \]

2. **Second Fraction**:
\[ \frac{(p - 1)(p + 1) \cdot 2p^2}{2p^2(p + 3)(p + 1)} \]

3. **Third Fraction**:
\[ \frac{p^2 \cdot 2p^2(p + 3)}{2p^2(p + 3)(p + 1)} \]

### Step 5: Combine the Fractions
Now we can combine the fractions:
\[ \frac{p(p - 1)(p + 1) + 2p^2(p - 1)(p + 1) + 2p^4(p + 3)}{2p^2(p + 3)(p + 1)} \]

### Step 6: Simplify the Numerator
1. Factor out common terms from the numerator:
\[ = \frac{(p - 1)(p + 1)(p + 2p^2) + 2p^4(p + 3)}{2p^2(p + 3)(p + 1)} \]

2. Combine like terms and simplify further.

### Step 7: Cancel Common Factors
After simplifying the numerator, cancel out any common factors with the denominator.

### Final Result
The simplest form of the expression is:
\[ \frac{1}{2p} \]