Set of values of $a$  for which the function $f:R\to R$  is given by $f\left(x\right)=x^3+(a+2)x^2+3ax+10$  is one-one, is given by

$f:R\to R;f\left(x\right)=x^3+(a+2)x^2+3ax+10$  is one-one.

Equational process in this is difficult, we use the fact that $f\left(x\right)$  is one-one in $R$ , then it must be either increasing in $R$  or decreasing in $R$

$\therefore f\text{'}\left(x\right)=3x^2+2(a+2)x+3a>0\forall x\in R$

                                                             $<0\forall x\in R$

As coefficient of $x^2=3>0$ , $f\text{'}\left(x\right)$  cannot be negative for all $x\in R$

$3x^2+2(a+2)x+3a>0\forall x\in R$

$\Rightarrow 4{(a+2)}^2-36a<0$

$\Rightarrow a^2-5a+4<0$  i.e., $(a-1)(a-4)<0$

$\Rightarrow 1<a<4$  i.e., $a\in (1,4)$ .