Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. -1,0,3+i. degree 4

Given Roots of the polynomial of degree 4 is -1, 0, 3+i
To find the polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Also, to Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.
Definition Used We know if a polynomial having a complex root then its complex conjugate will also be the root of the given polynomial.
Formation of the polynomial f(x)
Since by the definition if 3+i is one root then 3-i will also be the root of the given polynomial
In this way, we have a total number of 4 zeroes i.e.
-1,0,3+i and 3-i
Then the polynomial f(x) is
${\displaystyle f\left(x\right)=(x-(-1))(x-0)(x-(3+i)(x-(3-i))}$
${\displaystyle =x(x+1)(x-3-i)(x-3+i)}$
${\displaystyle =x(x+i)\{{(x-3)}^2-{\left(i\right)}^2\}}$
${\displaystyle =x(x+1)\{x^2-6x+9+1\}}$
${\displaystyle =(x^2+x)(x^2-6x+10)}$
${\displaystyle =x^4-6x^3+10x^2+x^3-6x^2+10x}$
${\displaystyle =x^4-5x^3+4x^2+10x}$