General term of $\left(a+\frac{b}{x^2}\right)$ is

$T_{r+1}={}^{11}C_r{\left(ax\right)}^{11-r}\times {\left(\frac{b}{x^2}\right)}^r$

$\Rightarrow T_{r+1}={}^{11}C_r a^{11-r}·b^r{\left(x\right)}^{11-3r}$

For coefficient of $x^{-7}$,

$11-3r=-7$

$\Rightarrow 3r=18$

$\Rightarrow r=6$

So, 

$T_{6+1}={}^{11}C_6 a^{11-6} b^6 x^{-7}$

$T_{r+1}=\left({}^{11}C_6 a^5{b}^6\right)x^{-7}$

So, $L={}^{11}C_6{a}^5{b}^6$

Similarly, general term of ${\left(b{x}^2+\frac{a}{x}\right)}^{11}$ is

$T_{r+1}={}^{11}C_r {\left(b{x}^2\right)}^{11-r}\times {\left(\frac{a}{x}\right)}^r$

$\Rightarrow T_{r+1}={}^{11}C_r b^{11-r}\times a^r\times {\left(x\right)}^{22-3r}$

So, for coefficient of $x^7$

 $22-3r=7 \Rightarrow r=5$

Hence,

$M{=}^{11}{C}_5{b}^6{a}^5$

So, $L+M=2\times {}^{11}C_6{a}^5{b}^6=924a^5{b}^6$

And, general term of ${\left(a{x}^2+\frac{b}{x}\right)}^{12}$ is

$T_{r+1}={}^{12}C_r{\left(a{x}^2\right)}^{12-r}{\left(\frac{b}{x}\right)}^r$

$T_{r+1}={}^{12}C_r a^{12-r} b^r x^{\left(24-3r\right)}$

For coeffiicent of $x^6$, $24-3r=6 \Rightarrow r=6$

So, coefficient of $x^6$ is

$={}^{12}C_6 a^6 b^6=924 a^6{b}^6$

Hence,

$L+M=\frac{1}{a}\times$[Coefficient of $x^6$ in ${\left(a{x}^2+\frac{b}{x}\right)}^{12}$]