Answer
$a.$ The value of the derivative is $f'(-2)=-14$.
$b.$ The equation is $y=-14x-16.$
Work Step by Step
$a.$ By definition of a derivative at point $a=-2$ we have
$$f'(-2)=\lim_{h\to0}\frac{f(-2+h)-f(-2)}{h}=\lim_{h\to0}\frac{4(-2+h)^2+2(-2+h)-(4(-2)^2+2(-2))}{h}=\lim_{h\to0}\frac{4(h^2+4-4h)+2h-4-12}{h}=\lim_{h\to0}\frac{4h^2+16-16h+2h-4-12}{h}=\lim_{h\to0}\frac{4h^2-14h}{h}.=\lim_{h\to0}(4h-14)=4\cdot0-14=-14$$
$b.$ We know that the equation of the tangent at point $(a,f(a))$ is given by $y-f(a)=f'(a)(x-a)$. Using $a=-2$, $f(a)=4(-2)^2+2(-2)=12$ and $f'(a)=-14$ we get
$$y-12=-14(x-(-2))\Rightarrow y-12=-14x-28$$ which gives $$y=-14x-16.$$
