Answer
$$\color{blue}{\bf{f(x)= 5 x^3 -10x^2+5x }}$$
Work Step by Step
We are given three zeros of a polynomial and the value of the function for a given ${x}$:
Our zeros are $\bf{ 0 }$ and $\bf{ 1 }$ of multiplicity 2 so let's make them into factors of our polynomial.
If $\bf{ 0 }$ is a zero then:
$( 0 +n)=0$
$n= 0 $
so our factor is $\bf{(x )}$ because when $x=\bf{ 0 }$, $(x +0 )=0$
If $\bf{ 1 }$ is a zero then:
$( 1 +n)=0$
$n= -1 $
so our factor is ${(x -1 )}$ because when $x=\bf{ 1 }$, $(x -1 )=0$
Since $\bf{ 1 }$ is a zero of multiplicity 2, $(x -1 )$ is a factor twice:
$\bf{(x -1 )^2}$ or $\bf{(x -1 )(x -1 )}$
Now we have three factors of our 3rd degree polynomial:
$\bf{(x )(x -1 )(x -1 )}$, which, multiplied by some unknown factor $a$, make up our function:
$f(x)=a(x )(x -1 )(x -1 )$
If $f( 2 )= 10 $ then
${(2 )(2 -1 )( 2 -1 )}$ times some number $a$, equals $ 10 $ or:
$f( 2 )= 10 =a (2 )(2 -1 )( 2 -1 ) $
$10 =a (2 )(1 )( 1 ) $
$10 =a (2 ) $
$\bf{a= 5 }$
Now that we have the value of $a$, we can find $f(x)$
$f(x)=5 (x )(x -1 )(x -1 ) $
$f(x)=5 (x^2 -x )(x -1 ) $
$f(x)=5 (x^3 -2x^2+x ) $
$$\color{blue}{\bf{f(x)= 5 x^3 -10x^2+5x }}$$
