#### Question

# Why does ((4x^2) + (12x) + 9) factor into (2x+3)(2x+3)?

can someone explain this to me?

#### Answer

((4x^2) + (12x) + 9)

First of all don't try to make it look complicated with unnecessary brackets and braces

write simply first 4x^2 + 12x + 9 is the one needs to be factorized

4x^2 + 12x + 9

= 4x^2 + 6x + 6x + 9

=2x(2x + 3) + 3(2x + 3)

= (2x + 3)(2x + 3) or (2x + 3)^2

If yo multiply out (2x+3)(2x+3) you get

2x * 2x = 4x^2

2x * 3 = 6x

3 * 2x = 6x

3 * 3 = 9

4x^2+12x+9=

(2x)^2+2(3)(2x)+3^2=

(2x+3)^2...........(formula: (a+b)^2=a^2+2ab+b^2)

(a+b)(a+b)=(a^2)+(2ab)+(b^2)

(4x^2)+(12x)+(9)

=((2x)^2)+(2(3)(2x))+(3^2)

=(2x+3)(2x+3)

(4x^2+12x+9) (2x+3)(2x+3)=(2x+3)^2 2x*2x=4x^2 2x*3=6x because you are squaring so you multiply by 2= 12x 3*3=9 4x^2+12x+9

((4x^2) + (12x) + 9) factors into (2x+3)(2x+3)

It is because ((4x^2) + (12x) + 9) = (2x)^2 + 2*2x*3 + 3^2 = (2x + 3)^2

Also a^2 + 2ab + b^2 = (a + b)^2 = (a + b)(a + b)

4x^2 + 12x + 9

36 Multiply a by c

Now, what multiplies to 36 and adds to 12?

= [(4x + 6)(4x + 6)]/4 Common factor

= [4(2x + 3)(2x + 3)]/4

= (2x + 3)^2

Why does ((4x^2) + (12x) + 9) factor into (2x+3)(2x+3)?

4x^2 + 12x + 9 is a perfect square trinomial.

When a trinomial is in the form

A^2 + 2AB + B^2

its factors are

(A + B)(A + B) = (A + B)^2

This is a special product and comes up quite frequently in Algebra. You should look in your textbook for "special products"

Practice.

QED

(2x + 3)^2