Answer
$21-36(x+2)+25(x+2)^2-8(x+2)^3+(x+2)^4$
Work Step by Step
Differentiate the given function $f(x)$ as follows:
$f'(x)=4x^3+2x \implies f'(-2)=-36 \\f''(x)=12x^2+2 \implies f''(-2)=50\\ f'''(x)=24x \implies f'''(2)=24$
The Taylor's series at $x=-2$ can be written as:
$f(x)=f(-2)+f'(-2) (x+2)+\dfrac{f''(-2)(x+2)^2 }{2!}+\dfrac{f''(-2) }{3!}(x+2)^3 +\dfrac{f''(-2) }{4!} (x+2)^4 \\=21-36(x+2)+25(x+2)^2-8(x+2)^3+(x+2)^4$
