We have,
$\alpha, \beta, \gamma$ are roots of equation
$$
\begin{aligned}
x^3+p_1 x^2+p_2 x+p_3 & =0 \\
S_r & =\alpha^\mu+\beta^r+\gamma^r \\
S_1 & =\alpha+\beta+\gamma=10 \\
S_2 & =\alpha^2+\beta^2+\gamma^2=38 \\
S_3 & =\alpha^3+\beta^3+\gamma^3=-1840
\end{aligned}
$$
Now, $\alpha^3+\beta^3+\gamma^3-3 \alpha \beta \gamma=(\alpha+\beta+\gamma)$
$$
\begin{array}{ll}
\left(\alpha^2+\beta^2+\gamma^2-(\alpha \beta+\beta \gamma+\gamma \alpha)\right) \\
\Rightarrow -1840+3 p_3=10 \\
\left.\left(38-\left[\frac{(\alpha+\beta+\gamma)^2-\left(\alpha^2+\beta^2+\gamma^2\right)}{2}\right]\right)\right)
\end{array}
$$


$$
\begin{array}{rlrl}
\Rightarrow & -1840+3 p_3 =10\left(38-\frac{\left(10^2-38\right)}{2}\right) \\
\Rightarrow & -1840+3 p_3 =5(76-100+38) \\
\Rightarrow & -1840+3 p_3 =70 \\
\Rightarrow & p_3 =\frac{1910}{3}
\end{array}
$$