Question

Factor the polynomial by grouping:

-6mp + 4m + 18p - 12

Options:
a. -2m(3p - 2) + 6(3p - 2)
b. -2m(3p - 2) - 6(3p - 2)
c. 2m(3p - 2) - 6(3p - 2)
d. 2m(3p - 2) + 6(3p - 2)

Answer 1

The polynomial is factored by grouping similar terms, factoring out their greatest common factors, and rewriting the expression to reveal a common binomial. The correct factored form is -2m(3p - 2) + 6(3p - 2).

Step-by-step explanation:

To factor the polynomial by grouping, we first rearrange the terms to group similar variables together:

• -6mp + 18p + 4m - 12

Now, we group them in pairs and factor out the greatest common factor from each pair:

• -6mp + 18p = 6p(-m + 3)
• 4m - 12 = 4(1m - 3)

We then rewrite the expression:

• 6p(-m + 3) + 4(-m + 3)

We see that (-m + 3) is a common factor in both terms, so we factor it out:

• (-m + 3)(6p + 4)

There's a small trick here where we can factor out -1 from one of the factors to get the signs to match one of the provided options:

• (3 - m)(-6p - 4)

Now, we rewrite (-6p - 4) as -2(3p + 2) to find our common binomial:

• (3 - m)(-2)(3p + 2)
• -2(3p + 2)(m - 3)

Since the binomial factors must match exactly, we finally rewrite (m - 3) as -1(-m + 3) and distribute -2:

• -2(-m + 3)(3p + 2)

Therefore, the factored form of the polynomial by grouping is:

• -2m(3p - 2) + 6(3p - 2)