For any real number $x$, if $ x^3+2x+33\neq 0$, then $x+3 \neq0$ How to solve this type of sum using indirect proof. Appreciate if anyone can explain it step by step.

Your statement is true. But we can even establish that the converse is true. 
Note that $$x^3+2x+33=(x+3)(x^2-3x+11)$$
But, since the polynomial discriminant is negative, for any real number, $x^2-3x+11 \neq 0$. 
So we have that if $x^3+2x+33=0 \Rightarrow x+3=0$. 
Also, since $(-3)^3+2(-3)+33=0$, we have that $x+3=0 \Rightarrow x^3+2x+33$. 
Thus, we have that $x^3+2x+33=0 \Leftrightarrow x+3=0$. 
This implies that $$x^3+2x+33 \neq 0 \Leftrightarrow x+3 \neq 0$$
