Given,
f(x)=(x2−2x+7)e(4x3−12x2−180x+31)
Now differentiating the function to find function is increasing or decreasing,
f′(x)=e(4x3−12x2−180x+31)(12(x2−2x−15)+2(x−1))
⇒f′(x)=e(4x3−12x2−180x+31)(12(x+3)(x−5)+2(x−1))
⇒f′(x)<0 for x∈[−3,0]
So, f(x)is decreasing function on [−3,0]
The absolute maximum value of the function f(x) is at x=−3
⇒α=−3