\(P=\dfrac{10x}{x^2+3x-4}-\dfrac{2x-3}{x+4}+\dfrac{x+1}{1-x}\) \(\left(x\ne1;x\ne-4\right)\)
\(P=\dfrac{10x}{\left(x+4\right)\left(x-1\right)}-\dfrac{2x-3}{x+4}-\dfrac{x+1}{x-1}\)
\(P=\dfrac{10x}{\left(x+4\right)\left(x-1\right)}-\dfrac{\left(2x-3\right)\left(x-1\right)}{\left(x+4\right)\left(x-1\right)}-\dfrac{\left(x+1\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)}\)
\(P=\dfrac{10x-2x^2+2x+3x-3-x^2-x-4x-4}{\left(x+4\right)\left(x-1\right)}\)
\(P=\dfrac{-3x^2+10x-7}{\left(x+4\right)\left(x-1\right)}\)
\(P=\dfrac{-3\left(x+1\right)\left(x-1\right)}{\left(x+4\right)\left(x-1\right)}\)
\(P=\dfrac{-3x-3}{x+4}\)
\(P=\dfrac{-3x-12+9}{x+4}\)
\(P=\dfrac{9}{x+4}-3\)
P nguyên khi \(\dfrac{9}{x+4}\) nguyên
\(\Rightarrow x+4\in\left\{1;-1;3;-3;9;-9\right\}\)
\(\Rightarrow x\in\left\{-3;-5;-1;-7;5;-13\right\}\)