[Cambridge/H2 Maths 2007] The complex number $w$ is such that $ww^* + 2w = 3 + 4i$, where $w^*$ is the complex conjugate of $w$. Find $w$ in the form $a + ib$, where $a$ and $b$ are real.

1. Substitute $w=a+ib$ into the given equation to express it in terms of $a$ and $b$.

Let $w = a + ib$, then the complex conjugate is $w^* = a - ib$.
$ww^* = (a+ib)(a-ib) = a^2 + b^2$
Substitute into $ww^* + 2w = 3 + 4i$:
$a^2 + b^2 + 2(a + ib) = 3 + 4i$
$a^2 + b^2 + 2a + 2bi = 3 + 4i$

2. Equate the real and imaginary parts to form simultaneous equations.

By comparing the real and imaginary parts on both sides:
Imaginary part:
$2b = 4 \implies b = 2$

Real part:
$a^2 + b^2 + 2a = 3$

3. Solve for $a$ by substituting the value of $b$.

Substitute $b=2$ into the real part equation:
$a^2 + (2)^2 + 2a = 3$
$a^2 + 2a + 4 = 3$
$a^2 + 2a + 1 = 0$
$(a + 1)^2 = 0 \implies a = -1$

4. State the final complex number $w$.

Therefore, the complex number is:
$w = -1 + 2i$