Designing a PD controller for a unity feedback control system
In the equation \$(s+1)(s+2)+3(K_P+sK_D)=0\$, substitute \$s=-4+4 i\$ and \$s=-4-4 i\$.

This will give you two equations in the two unknowns \$K_P\$ and \$K_D\$.

The solution I am getting after this is \$K_P=10\$ and \$K_D=\frac{5}{3}\$.