<s>[INST] 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.
 [/INST] 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 $$x^3+2x+33 \neq 0 \Leftrightarrow x+3 \neq 0$$
</s>