Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

Question

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?

Answer 1

$$x_1\overline {x_1}-i\overline{x_1} x_2+ix_1\overline {x_2}+2x_2\overline {x_2}=(ix_1+x_2)\overline{(ix_1+x_2)}+x_2\overline {x_2}=\|ix_1+x_2\|^2+\|x_2\|^2$$