`pq ^(2) + q (p -1) -1=` ?

Options

A. `(pq+1) (q-1)`
B. `p (1 +1)(q-1)`
C. `q(p-1)( q+1)`
D. `(pq-1) (q+1)`

Answer

To factor the expression \( pq^2 + q(p - 1) - 1 \), we will follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ pq^2 + q(p - 1) - 1 \]

### Step 2: Distribute the term \( q(p - 1) \)
Distributing \( q \) in the term \( q(p - 1) \):
\[ pq^2 + qp - q - 1 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ pq^2 + qp - q - 1 = pq^2 + qp - (q + 1) \]

### Step 4: Factor out common terms
Notice that \( pq^2 + qp \) has a common factor of \( q \):
\[ q(pq + p) - (q + 1) \]

### Step 5: Factor further if possible
We can see if we can factor \( q(p + 1) - 1 \):
\[ q(p + 1) - 1 \]

### Step 6: Final expression
Thus, the expression can be factored as:
\[ q(p + 1) - 1 \]