ANSWER
The zeros are:
[tex]x = 0 \: \: or \: \: x = - 5[/tex]
with multiplicity of 4 and 2 respectively.
EXPLANATION
The given function is
[tex]f(x) =3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4} [/tex]
We want to find
[tex]3 {x}^{6} + 30 {x}^{5} + 75 {x}^{4} = 0 [/tex]
We can find the zeros of this function by factorizing the greatest common factor to obtain,
[tex]3 {x}^{4} ( {x}^{2} + 10x + 25) = 0[/tex]
The expression in the bracket can be rewritten as,
[tex]3 {x}^{4} ( {x}^{2} + 2(5)x + {5}^{2} ) = 0[/tex]
We can see clearly that, the expression in the bracket is a perfect square that can be factored as,
[tex]3 {x}^{4} ( x+ 5)^2 = 0[/tex]
This implies that,
[tex]3 {x}^{4} = 0[/tex]
or
[tex](x + 5) ^{2} = 0[/tex]
This gives,
[tex] {x}^{4} = 0 \: \: or \: \: x + 5 = 0[/tex]
This finally gives,
[tex]x = 0 \: \: or \: \: x = - 5[/tex]
