Can someone give me a hint to get me started? Below is the problem. It's proving that a function is equal to a big O function.
$f \sim \mathcal{O}(g) \Leftrightarrow \exists M >0,x_0 \ni x>x_0 \Rightarrow |f| \leq M|g|$
here
$f(n) = 4n^4 + 5n^2 +32$
$g(n) = n^4$
we need to find $M>0,~n_0$
such that
$n>n_0 \Rightarrow | 4n^4 + 5n^2 +32| \leq M |n^4|$
everything is positive so we can drop the absolute value signs
$n>n_0 \Rightarrow (4-M)n^4 + 5n^2 + 32 \leq 0$
$n_0=1,~M=41$ satisfies this inequality, so does
$n_0=3,~M=5$