Determine algebraically whether the function is even, odd, or neither. Give reasons for your answer. f(x)=(3x+2)^2 (x-4)(x+1)(2x-3)

`f(x) =
(3x+2)^2(x-4)(x+1)(2x-3)`

A function is even if `f(-x) =
f(x)` for all x   An example would be f(x)=x^4

A function
is odd if `f(-x) = -f(x)` for all x   An example would be
f(x)=x^3

If neither one of these is true then the function
is neither even nor odd.  An example would be f(x) = 2x +
1

So we want to calculate
f(-x)

`f(-x) = (-3x+2)^2 (-x-4)(-x+1)(-2x-3)` Now we
simplify

`f(-x) = ((-1)(3x - 2))^2
((-1)(x+4))((-1)(x-1))((-1)(2x+3)`

`f(-x) = (-1)^2
(-1)(-1)(-1) (3x - 2)^2 (x+4)(x-1)(2x+3)`

`f(-x) =
-(3x-2)^2 (x+4)(x-1)(2x+3)`

We can see that f(-x) `!=` f(x)
and f(-x) `!=` -f(x)

So the function is neither even or
odd.