Can someone show me how to solve these step-by-step? Please? Thank you.

Factor each polynomial by grouping. Check your answer.

1. 2r^2 - 6r + 12 - 4r
2. 14q^2 - 21q + 6 - 4q
3. w^3 -4w^2 + w - 4
4. 2p^3 - 6p^2 + 15 - 5p

1.

2r(r - 3) + 4(3 - r) , ahhh, did you notice that the brackets are opposite? , so ..
= 2r(r-3) - 4(r - 3)
= (r-3)(2r-4)

Use the same "trick" on the others if you have to.

My textbook says the answer is 2(r-2)(r-3), though.

Surely you could see that 2 was an additional common factor contained in my (2r-4) factor.

What? I don't understand what you mean.

look at my answer ...

(r-3)(2r-4)
= (r-3)(2)(r-2)
= 2(r-3)(r-2)

Oh.

My answers to the other equations:

2. ?
3. (w^2 + 1)(w - 4)
4. (2p^2 - 5)(p - 3)

2. (7q - 2)(2q - 3).

3 and 4 are correct

#2

14q^2 - 21q + 6 - 4q
= 7q(2q - 3) + 2(3-2q)
= 7q(2q-3) - 2(2q-3)
= (2q-3)(7q-2)

Thank you.

Of course! I'd be happy to help you solve these polynomial problems step-by-step.

1. 2r^2 - 6r + 12 - 4r
To factor by grouping, we group the terms in pairs:
(2r^2 - 6r) + (12 - 4r)

Now, we factor out the greatest common factor from each pair separately:
2r(r - 3) - 4(r - 3)

Now we have a common binomial factor, (r - 3). We can factor it out:
(r - 3)(2r - 4)

You can check your answer by distributing the factored form and seeing if it simplifies back to the original polynomial.

2. 14q^2 - 21q + 6 - 4q
Again, let's group the terms:
(14q^2 - 21q) + (6 - 4q)

Factor out the greatest common factor from each pair:
7q(2q - 3) - 2(2q - 3)

Now we have a common binomial factor, (2q - 3). Let's factor it out:
(2q - 3)(7q - 2)

To check, distribute the factored form to see if it simplifies back to the original polynomial.

3. w^3 - 4w^2 + w - 4
Group the terms:
(w^3 - 4w^2) + (w - 4)

Factor out the greatest common factor from each group:
w^2(w - 4) + (w - 4)

Now, we notice that we have a common binomial factor, (w - 4). Let's factor it out:
(w - 4)(w^2 + 1)

Check by distributing and simplifying the factored form.

4. 2p^3 - 6p^2 + 15 - 5p
Group the terms:
(2p^3 - 6p^2) + (15 - 5p)

Factor the greatest common factor from each group:
2p^2(p - 3) + 5(3 - p)

Rearrange the terms to match the order of the binomials:
2p^2(p - 3) - 5(p - 3)

Now we have a common binomial factor, (p - 3). Let's factor it out:
(p - 3)(2p^2 - 5)

You can always check your answer by distributing the factored form and simplifying it back to the original polynomial.