If _ x → 0 a x ^ 2 e ^ x - b _ e (1 + x) + cx e ^ - x x ^ 2 x = 1 ,...

Given: 

$\lim_{x \to 0}\dfrac{a{x}^{2}{e}^{x} - b{\log}_{e}\left(1 + x\right) + cx{e}^{- x}}{{x}^{2}\sin x} = 1$

$⇒ \lim_{x \to 0}\dfrac{a{x}^{2}{e}^{x} - b{\log}_{e}\left(1 + x\right) + cx{e}^{- x}}{{x}^{2} \cdot x \cdot \dfrac{\sin x}{x}} = 1$

$⇒ \lim_{x \to 0}\dfrac{a{x}^{2}\left(1 + \dfrac{x}{1 !} + \dfrac{{x}^{2}}{2 !} + . . .\right) - b\left(x - \dfrac{{x}^{2}}{2} + \dfrac{{x}^{3}}{3} + . . .\right) + cx\left(1 - \dfrac{x}{1 !} + \dfrac{{x}^{2}}{2 !} + . . .\right)}{{x}^{2} \times x} = 1$

Coefficient of $x = 0$

$⇒ - b + c = 0 . . . \left(iii\right)$

Coefficient of ${x}^{2} = 0$

$⇒ a + \dfrac{b}{2} - c = 0 . . . \left(ii\right)$

$⇒ a - \dfrac{c}{2} = 0$

$⇒ a = \dfrac{c}{2}$

Coefficient of ${x}^{3} = 1$

$⇒ a - \dfrac{b}{3} + \dfrac{c}{2} = 1 . . . \left(iii\right)$

$⇒ \dfrac{c}{2} - \dfrac{c}{3} + \dfrac{c}{2} = 1$

$⇒ \dfrac{2c}{3} = 1$

$⇒ c = \dfrac{3}{2} = b$

$⇒ a = \dfrac{3}{4}$

$⇒ 16\left({a}^{2} + {b}^{2} + {c}^{2}\right) = 16\left(\dfrac{9}{16} + \dfrac{9}{4} \times 2\right)$

$⇒ 16\left({a}^{2} + {b}^{2} + {c}^{2}\right) = 81$