Let (a + bx + c x ^ 2 ) ^ 10 = ∑ _ i = 10 ^ 20 p _ i x ^ i , a, b,...

Given that ${\left(a + bx + c{x}^{2}\right)}^{10} = \mathop{∑}_{i = 10}^{20}{p}_{i}{x}^{i}, a, b, c \in \mathbb{N}$

General term : $\dfrac{10 !}{{r}_{1} ! {r}_{2} ! {r}_{3} !}{\left(a\right)}^{{r}_{1}}{\left(bx\right)}^{{r}_{2}}{\left(c{x}^{2}\right)}^{{r}_{3}}$

For coefficient. of $x: {r}_{2} + 2{r}_{3} = 1$

Let $\begin{cases}{r}_{1} & {r}_{2} & {r}_{3} \\ 9 & 1 & 0\end{cases}$

$∴$ coefficient of $x = \dfrac{10 !}{9 !}{a}^{9}{b}^{1} = 20$

$⇒ {a}^{9} \cdot b = 2 . . . . . . \left(i\right)$

Coefficient of $⇒ {x}^{2}: \dfrac{10 !}{8 ! 2 ! 0 !}{a}^{8} \cdot {b}^{2} + \dfrac{10 !}{9 ! 0 ! 1 !} \cdot {a}^{9} \cdot c = 210$

$⇒ 45{a}^{8} \cdot {b}^{2} + 10 \cdot {a}^{9} \cdot c = 210$

$⇒ 9{a}^{8}{b}^{2} + 2{a}^{9} \cdot c = 42 . . . . . . . \left(ii\right)</p><p>As$a, b, c \in N, so by hit and $trial in equation \left(i\right) \text{\text{ and } }$ \left(ii\right)$we get,$a = 1, b = 2, c = 3$</p><p>Hence,$2\left(a + b + c\right) = 2 \left(3 + 2 + 1\right) = 12