2007 H2 Mathematics

Out of syllabus (complex locus).

The complex number $w$ is such that

$$w{w}^{*}+2w=3+4\mathrm{i},$$

where $w^{*}$ is the complex conjugate of $w.$ Find $w$ in the form $a+\mathrm{i}b,$ where $a$ and $b$ are real.