If $x^4+4x^3-2x^2-12x+9=0$  has a pair of equal roots then the roots are

$1+4-2-12+9=0$

Sum of the coefficients is 0, then 1 is a root

Given roots are in pair of equal roots

By substituting $-3$  in $f\left(x\right)$

$={\left(-3\right)}^4+4{\left(-3\right)}^3-2{\left(-3\right)}^2-12\left(-3\right)+9$

$=81+4\left(-27\right)-2\left(9\right)+36+9$

$=81-108-18+36+9$

$=0$

$\therefore$ The roots are $1,1,-3,-3$