Fantastic four

Put $a=b=0$ in the equation: $f(0)=2f(0)+0\implies \color{#D61F06}{f(0)=0}$

Put $a=x,b=-x$ in the equation:

$\color{#D61F06}{\underbrace{f(0)}_0}=f(x)+f(-x)-4x^2$ $\implies f(x)=4x^2-f(-x)~~(\small{\color{grey}{Generalisation}})$

Put $x=4$ so that:

$f(4)=4(4)^2-40=\large{\boxed{24}}$

$f(a+b)=f(a)+f(b)+4ab\\\Rightarrow f(a)=f(a)+f(0)\\\Rightarrow f(0)=0\\\Rightarrow f(-4+4)=0\\\Rightarrow f(-4)+f(4)-64=0\Rightarrow40-64+f(4)=0\Rightarrow f(4)=24$

$\begin{aligned} f(-4) & = 40 \\ f(-2-2) & = 40 \\ f(-2)+f(-2)+4(-2)(-2) & = 40 \\ 2f(-2) + 16 & = 40 \\ 2f(-2) & = 24 \\ \implies f(-2) & = 12 \end{aligned}$

$\begin{aligned} f(4-2) & = f(4) + f(-2) + 4(4)(-2) \\ f(2) & = f(4) + 12 - 32 \\ \implies f(4) & = f(2) + 20 \quad ...(1) \end{aligned}$

$\begin{aligned} f(2+2) & = f(2) + f(2) + 4(2)(2) \\ \implies f(4) & =2 f(2) + 16 \quad ...(2) \end{aligned}$

$\begin{aligned} 2 \times (1)-(2): \quad f(4) & = 40 - 16 = \boxed{24} \end{aligned}$