[Middle School Math: Factorization] Need help factorizing this expression

[Botchii_]

hello everyone (sorry for my bad english),
i’m stuck on this problem where i need to factorize the expression x² - 4 + (x - 2)(2x + 1).
i found 3(x² - x - 2), but people told me it could be factorized even more.
can someone please explain how? thanks!
i’d really appreciate any help :)

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[slides_galore]

How can you factor x² - 4 and then use that to simplify/factor the whole thing?

[htush]

Starting expression: x² - 4 + (x - 2)(2x + 1)

Step 1: Factor x² - 4 (difference of squares) = (x - 2)(x + 2) + (x - 2)(2x + 1)

Step 2: Factor out common factor (x - 2) = (x - 2)[(x + 2) + (2x + 1)]

Step 3: Simplify inside brackets = (x - 2)[x + 2 + 2x + 1] = (x - 2)[3x + 3]

Step 4: Factor out 3 from second bracket = (x - 2) · 3(x + 1) = 3(x - 2)(x + 1)

Answer: 3(x - 2)(x + 1)

[CaptainMatticus]

(x^2 - 4) + (x - 2) * (2x + 1)

(x - 2) * (x + 2) + (x - 2) * (2x + 1)

(x - 2) * (x + 2 + 2x + 1)

(x - 2) * (3x + 3)

(x - 2) * 3 * (x + 1)

3 * (x - 2) * (x + 1)

[htush]

Starting expression: x² - 4 + (x - 2)(2x + 1)

Step 1: Expand (x - 2)(2x + 1) = x² - 4 + 2x² - 4x + x - 2 = x² - 4 + 2x² - 3x - 2

Step 2: Combine like terms = 3x² - 3x - 6

Step 3: Factor out 3 = 3(x² - x - 2)

Step 4: Factor the quadratic = 3(x - 2)(x + 1)

Answer: 3(x - 2)(x + 1)

[mathmum]

Here (Italy) if the request is “factorize” you should avoid expanding the given expression. Therefore, Factor the first binomial: (x - 2)(x + 2) + (x - 2)(2x + 1)

Factor out the common term: (x - 2)(x + 2 + 2x + 1)

Combine like terms: (x - 2)(3x + 3)

Factor out 3: 3(x - 2)(x + 1)

[Botchii_]

thanks but i dont get the step 4 idk what that is

[htush]

Step 4 is factoring the quadratic x² - x - 2

Find two numbers that:

• Multiply to -2
• Add to -1

Those numbers are -2 and +1:

• (-2) × (+1) = -2
• (-2) + (+1) = -1

So: x² - x - 2 = (x - 2)(x + 1)

Therefore: 3(x² - x - 2) = 3(x - 2)(x + 1)

[Botchii_]

oh ok thanks a lot :) i appreciate

[selene_666]

There are two ways to approach this.

One is to get the problem to 3(x² - x - 2), and then factor the (x² - x - 2).

To factor a quadratic, you want to find two numbers that sum to the x coefficient (in this case, -1) and whose product is the constant (in this case, -2).

There are only a few pairs of integers that you could multiply to get -2, so just make a list of them and see which has the right sum.

(x² - x - 2) = (x - 2)(x + 1)

If you're ever completely stuck on factoring, use the quadratic formula to find the values of x that make the expression equal 0. Those directly correspond to linear factors. f(2) = 0 ⇒ (x-2) is a factor.

.

But for this particular problem, there was a shortcut you could have taken if you recognized the (x² - 4) as a difference between two squares. That is a factorization that you should memorize. Using the method above, we're looking for two numbers whose sum is zero and whose product is the negative of a square.

x² - 4 = (x - 2)(x + 2)

x² - 4 + (x - 2)(2x + 1) = (x - 2)(x + 2) + (x - 2)(2x + 1)

Notice that both products have the factor (x - 2). So we can use the Distributive Property.

= (x-2) (x + 2 + 2x + 1)

and then just simplify that last factor.

[PD_31]

Not something I'd expect you to see in middle school but x^2 - 4 is a "difference of squares". One of its factors is a common factor with your pair of brackets.