[tex3]P(x)\text{ polinômio com }P(17)=10\text{ e }P(24)=17\\
n_1,n_2 \text{ raízes inteiras de }P(x)\!\!+\!\!x\!\!+\!\!3\\[24pt]
\text{Seja }\nu \in\{n_1;n_2\}:\\
\exists k\in\mathbb{Z}/P(\nu)-P(24)=(\nu-24)k\quad\quad\small\left(\forall x_0,x_1\!\in\!\mathbb{R},P(x_1)\!-\!P(x_0)\!=\!\!\sum_{k=0}^{n}a_k(x_1^k\!\!-\!\!x_o^k)\!=\!(x_1\!-\!x_0)\sum_{k=0}^n \!a_k\!\sum_{i=0}^kx_1^{k-i}x_oî\right)\text{ e aqui }x_0,x_1,a_k\!\in\!\mathbb{Z}\\
\text{e }\begin{array}[t]{rl}
P(\nu)-P(24)=(\nu-24)k&\implies\nu+3-17=(\nu-24)k\\
&\implies\nu(k-1)=-14+24k\\
&\implies\nu k'=-14+24+24k'\quad\quad \text{com }k'=k-1\\
&\implies \nu=\dfrac{10}{k'}+24=\dfrac{1\times2\times5}{k'}+24\\
&\implies k'\in\{-10;-5;-2;-1;1;2;5;10\}\quad\quad\text{já que }\nu\in\mathbb{Z}\\
&\implies \nu\in\{14;19;22;23;25;26;29;34\}
\end{array}\\[96pt]
\exists k\in\mathbb{Z}/P(\nu)-P(17)=(\nu-17)k\\
\text{e }\begin{array}[t]{rl}
P(\nu)-P(17)=(\nu-17)k&\implies\nu+3-10=(\nu-17)k\\
&\implies\nu(k-1)=-7+17k\\
&\implies\nu k'=-7+17+17k'\quad\quad \text{com }k'=k-1\\
&\implies \nu=\dfrac{10}{k'}+17=\dfrac{1\times2\times5}{k'}+17\\
&\implies k'\in\{-10;-5;-2;-1;1;2;5;10\}\quad\quad\text{já que }\nu\in\mathbb{Z}\\
&\implies \nu\in\{7;12;15;16;18;19;22;27\}
\end{array}\\[96pt]
\text{e então }(n_1,n_2)=(19,22)\text{ ou }(n_1,n_2)=(22,19)\text{ já que }n_1\neq n_2\\[36pt][/tex3]
[tex3]
\boxed{\hspace{0.5cm}\\[6pt]\hspace{0.5cm}n_1\cdot n_2=19\times 22=418\hspace{0.5cm}\\\hspace{0.5cm}}
[/tex3]