Formula use:

a^3+b^3=\left(a+b\right)^3-3.a.b\left(a+b\right)

Explanation

According to the question,
2x+\large\frac{2}{x}\normalsize=3
\rightarrow2\left(x+\large\frac{1}{x}\right)\normalsize=3
\therefore x+\large\frac{1}{x}\normalsize=\large\frac{3}{2}
Here, given expression is :
x^3+\large\frac{1}{x^3}\normalsize + 2
\rightarrow\{\left(x\right)^3+\large\left(\frac{1}{x}\right)\normalsize^3\}+2
\rightarrow\{\left(x+\large\frac{1}{x}\right)^3\normalsize-3.x.\large\frac{1}{x}\normalsize\left(x+\large\frac{1}{x}\right)\normalsize\}+2
Now, x+\large\frac{1}{x}\normalsize=\large\frac{3}{2}
\therefore\{\large\left(\frac{3}{2}\right)^3\normalsize-3\times1\times\large\frac{3}{2}\normalsize\}+2
\rightarrow\large\left(\frac{27}{8}-\frac{9}{2}\right)\normalsize+2
\rightarrow\large\frac{27-36+16}{8}
\rightarrow\large\frac{43-36}{8}
\therefore\large\frac{7}{8}
Ans: \frac{7}{8} .