1. Replace the asterisk (*) in the expression $\left(x^{4}-3\right)^{2}+\left(x^{3}+*\right)^{2}$ with a monomial so that after squaring and combining like terms, there are four terms.
Solution. $\quad$ Replace the asterisk (*) with $3 x: \quad\left(x^{4}-3\right)^{2}+\left(x^{3}+3 x\right)^{2}=$ $x^{8}-6 x^{4}+9+x^{6}+6 x^{4}+9 x^{2}=x^{8}+x^{6}+9 x^{2}+9$