To factor the polynomial 2p - 6q + pq - 3q^2 completely, we can follow these steps:
Step 1: Grouping
Since we have four terms, we can try grouping the terms in pairs. Let's group the first two terms and last two terms together:
(2p - 6q) + (pq - 3q^2)
Step 2: Factoring out common factors
Now, let's look for common factors in each group. The first group (2p - 6q) has a common factor of 2, so we can factor it out:
2(p - 3q)
The second group (pq - 3q^2) has a common factor of q, so we can factor it out:
q(p - 3q)
Step 3: Common factor (p - 3q)
Now that we have the common factor (p - 3q) in both groups, we can rewrite the expression:
2(p - 3q) + q(p - 3q)
Step 4: Factor the common binomial (p - 3q)
The expression now has a common binomial factor of (p - 3q). We can factor it out:
(p - 3q)(2 + q)
Therefore, the polynomial 2p - 6q + pq - 3q^2 is completely factored as (p - 3q)(2 + q).
Remember, the first step in factoring a polynomial is to try grouping the terms. Then, look for common factors in each group, and finally factor out any common binomial factors.