Given that: $f\left(x\right)=x^6+2x^4+x^3+2x+3, x\in \mathrm{R}$

So, $f\left(1\right)=9$

$\underset{x\to 1}{\lim }\frac{9{\mathrm{x}}^{\mathrm{n}}-x^6-2{\mathrm{x}}^4-x^3-2\mathrm{x}-3}{x-1}=44$

$\Rightarrow \underset{x\to 1}{\lim }\frac{9{\mathrm{x}}^{\mathrm{n}}-9-x^6+1-2{\mathrm{x}}^4+2-x^3+1-2\mathrm{x}+2}{x-1}=44$

$\Rightarrow \underset{x\to 1}{\lim }9\left(\frac{x^n-1}{x-1}\right)-\left(\frac{x^6-1}{x-1}\right)-2\left(\frac{x^4-1}{x-1}\right)-\left(\frac{x^3-1}{x-1}\right)-2\left(\frac{x-1}{x-1}\right)=44$

As we know that, $\underset{x\to a}{\lim }\left(\frac{x^n-a^n}{x-a}\right)=n{a}^{n-1}$

$\Rightarrow 9\mathrm{n}-6-8-3-2=44$

$\Rightarrow n=7$