Answer
$-7+23(x+1)-41(x+2)^2+36(x+1)^3-16(x+1)^4+3(x+1)^5$
Work Step by Step
Differentiate the given function $f(x)$ as follows:
$f'(x)=15x^4-4x^3+6x^2+2x \implies f'(-1)=23 \\ f''(x)=60x^3-12x^2+12x+2 \implies f''(-1)=-82\\ f'''(x)=180x^2-24x+12 \implies f'''(-1)=216$
The Taylor series at $x=-1$ is as follows:
$f(x)=f(-1)+f'(-1) (x+1)+\dfrac{f''(-1)}{2!}(x+1)^2 +\dfrac{f''(-1)}{3!} (x+1)^3 +\dfrac{f''(-1)}{4!} (x+1)^4 \\=-7+23(x+1)-41(x+2)^2+36(x+1)^3-16(x+1)^4+3(x+1)^5$
