rational expressions m3 -7/m2 -4

To simplify the rational expression (m^3 - 7) / (m^2 - 4), we can factor both the numerator and denominator.

First, let's start with factoring the numerator, (m^3 - 7):

We can rewrite it as (m^3 - (2^3)) to see if it fits the form of difference of cubes.
Using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2), we have:

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

Now, let's factor the denominator, (m^2 - 4):

This can be recognized as the difference of squares, which can be factored as (a^2 - b^2) = (a - b)(a + b):

(m^2 - 4)
= (m^2 - 2^2)
= (m - 2)(m + 2)

Now that we have factored both the numerator and denominator, we can simplify the rational expression:

(m^3 - 7) / (m^2 - 4)
= [(m - 2)(m^2 + 2m + 4)] / [(m - 2)(m + 2)]

Notice that the (m - 2) terms in both the numerator and denominator cancel out, simplifying the expression further:

= (m^2 + 2m + 4) / (m + 2)

So, the simplified form of the rational expression (m^3 - 7) / (m^2 - 4) is (m^2 + 2m + 4) / (m + 2).