3p-2 p-3
__________ - _______
p^2+2p-24 P^2-16
subtract & simplify
To subtract and simplify the expression, we need to find a common denominator for the two fractions. In this case, the common denominator is (p^2 + 2p - 24)(p^2 - 16).
Let's simplify each fraction first:
3p - 2 / (p^2 + 2p - 24) can be factored as (3p - 2) / (p + 6)(p - 4).
p - 3 / (p^2 - 16) can be factored as (p - 3) / (p + 4)(p - 4).
Now, we can write the expression as a single fraction:
[(3p - 2)(p + 4)(p - 4) - (p - 3)(p + 6)(p - 4)] / [(p + 6)(p - 4)(p + 4)(p - 4)].
Next, expand and simplify the numerator:
(3p - 2)(p + 4)(p - 4) - (p - 3)(p + 6)(p - 4) can be simplified as:
[3p(p^2 - 16) + 12p(p - 4) - 2(p - 4) - p(p^2 + 12p - 24) + 3(p^2 + 6p - 24)].
After simplification, the numerator becomes:
[3p^3 - 48p + 12p^2 - 24p - 2p + 8 - p^3 - 12p^2 + 24p + 3p^2 + 18p - 72].
Now, combine like terms in the numerator:
[2p^3 + 3p^2 - 46p - 64].
The denominator is already factored: [(p + 6)(p - 4)(p + 4)(p - 4)].
So, the simplified expression is:
(2p^3 + 3p^2 - 46p - 64) / [(p + 6)(p - 4)(p + 4)(p - 4)].