(b) $$ 3 p^{2}+7 p q-6 q^{2} $$

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UpStudy AI · Accepted Answer

1. Start with the expression:
   $$
   3p^2 + 7pq - 6q^2
   $$

2. To factor, consider splitting the middle term. Look for two numbers that multiply to $$ 3 \times (-6q^2) = -18q^2 $$ and add to $$ 7q $$. The numbers $$ 9q $$ and $$ -2q $$ satisfy this since:
   $$
   9q + (-2q) = 7q \quad \text{and} \quad 9q \times (-2q) = -18q^2.
   $$

3. Rewrite the middle term using these numbers:
   $$
   3p^2 + 9pq - 2pq - 6q^2.
   $$

4. Group the terms:
   $$
   (3p^2 + 9pq) + (-2pq - 6q^2).
   $$

5. Factor out the common factors from each group:
   $$
   3p(p + 3q) - 2q(p + 3q).
   $$

6. Factor out the common binomial $$(p + 3q)$$:
   $$
   (p + 3q)(3p - 2q).
   $$

Thus, the factored form of $$ 3p^2 + 7pq - 6q^2 $$ is:
$$
(p + 3q)(3p - 2q)
$$
$$ 3p^2 + 7pq - 6q^2 = (p + 3q)(3p - 2q) $$

(b) \( 3 p^{2}+7 p q-6 q^{2} \)

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