Simplify (27x^3-64)/(9x^2-16)

To evaluate the expression we'll use factorization. We
notice that the numerator is a difference of
cubes:

27x^3-64 = (3x)^3 -
(4)^3

We'll apply the
formula:

a^3 - b^3 = (a-b)(a^2 + ab +
b^2)

We'll put a = 3x and b =
4

(3x)^3 - (4)^3 = (3x-4)(9x^2 + 12x +
16)

We also notice that the denominator is a difference of
squares:

9x^2-16 = (3x)^2 -
4^2

We'll apply the
formula:

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

(3x)^2 - 4^2 =
(3x-4)(3x+4)

We'll substitute the differences by their
products:

 [(27x^3-64)/(9x^2-16)] = (3x-4)(9x^2 + 12x +
16)/(3x-4)(3x+4)

We'll simplify by the common factor
(3x-4):

 [(27x^3-64)/(9x^2-16)] =  [(9x^2 +
12x + 16)/(3x+4)]

We can also combine the
terms 12x + 16 and factorize them by
4;

 [(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) + (12x +
16)/(3x+4)

 [(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) + 4(3x +
4)/(3x+4)

We'll simplify the last ratio by (3x+4) and we'll
get:

 [(27x^3-64)/(9x^2-16)]  = 9x^2/(3x+4) +
4