2a2-a-34a2-9

For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=2⋅-3=-6 and whose sum is b=-1.

Factor -1 out of -a.

2a2-(a)-34a2-9

Rewrite -1 as 2 plus -3

2a2+(2-3)a-34a2-9

Apply the distributive property.

2a2+2a-3a-34a2-9

2a2+2a-3a-34a2-9

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

(2a2+2a)-3a-34a2-9

Factor out the greatest common factor (GCF) from each group.

2a(a+1)-3(a+1)4a2-9

2a(a+1)-3(a+1)4a2-9

Factor the polynomial by factoring out the greatest common factor, a+1.

(a+1)(2a-3)4a2-9

(a+1)(2a-3)4a2-9

Rewrite 4a2 as (2a)2.

(a+1)(2a-3)(2a)2-9

Rewrite 9 as 32.

(a+1)(2a-3)(2a)2-32

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2a and b=3.

(a+1)(2a-3)(2a+3)(2a-3)

(a+1)(2a-3)(2a+3)(2a-3)

Cancel the common factor.

(a+1)(2a-3)(2a+3)(2a-3)

Rewrite the expression.

a+12a+3

a+12a+3

Simplify (2a^2-a-3)/(4a^2-9)