Question

How Could You Factor This

2(a+b)^2+5(a+b)+3

Answer 1

Replace (a + b) with x.
Does this look familiar now?

Answer 2

Hmm, Ill try that

Answer 3

2x^2+5x+3

Answer 4

@mathstudent55

What can I do from here

Answer 5

Do you know how to factor that?
You need two sets of parentheses:
( )( )

Answer 6

Yea but there is 5x and what combination adds to 5 that also has a product of 6?

Answer 7

The numbers here are small so we can use the guess method.
The 2x^2 term is 2x * x, right?
So it goes here: (2x )(x )

Answer 8

Is there a mathematical way rather than guessing?

Answer 9

The term 3 is 3 * 1, so now you need to look at only two possibilities:
(2x + 3)(x + 1)
(2x + 1)(x + 3)

Answer 10

Yes, I can show you the mathematical way also.

Answer 11

Yea, I would like to see the mathematical way

Answer 12

How can we factor

2x^2+5x+3

Answer 13

Let's start from the beginning.

We are factoring a trinomial of the form
\(ax^2 + bx + c\)

Your trinomial is
\(2x^2 + 5x + 3\)

You have a = 2, b = 5, and c = 3, ok?

Answer 14

1. Multiply ac together.
For you, ac = 2*3 = 6.

Answer 15

yea, just got that

Answer 16

2. Find two factors of ac whose sum is b.
In your case, you need two factors of 6 whose sum is 5.

Answer 17

Yep and the final answer which I got was

(2a+2b+3)(a+b+1)

Answer 18

3. Break up the middle term with the two factors you found.
In your case, the factors are 2 and 3.
Now you write 2x^2 + 2x + 3x + 3

4. Factor by parts. That is, factor a common factor out of the first two terms amd a common factor out of the last two terms.
In your case, 2x(x + 1) + 3(x + 1)

5. Factor out the common term.
In your case:
(x + 1)(2x + 3)

Answer 19

Now we substitute back the a + b for x that we did earlier on.

(x + 1)(2x + 3)
= (a + b + 1)[2(a + b) + 3]
= (a + b + 1)(2a + 2b +3)

Which is what you got.

Good job!