P=\(\frac{a^3-4a^2-a+4}{a^3-7a^2+14a-8}=\frac{\left(a^3-4a^2\right)-\left(a-4\right)}{\left(a^3-8\right)-\left(7a^2-14a\right)}\)

\(=\frac{a^2\left(a-4\right)-\left(a-4\right)}{\left(a-2\right)\left(a^2+2a+4\right)-7a\left(a-2\right)}\)

\(=\frac{\left(a-4\right)\left(a^2-1\right)}{\left(a-2\right)\left(a^2-5a+4\right)}\)

\(=\frac{\left(a-4\right)\left(a^2-1\right)}{\left(a-2\right)\left(\left(a^2-4a\right)-\left(a-4\right)\right)}\)

\(=\frac{\left(a-4\right)\left(a-1\right)\left(a+1\right)}{\left(a-2\right)\left(a\left(a-4\right)-\left(a-4\right)\right)}\)

\(=\frac{\left(a-4\right)\left(a-1\right)\left(a+1\right)}{\left(a-2\right)\left(a-4\right)\left(a-1\right)}\)

\(=\frac{a+1}{a-2}\)

Chúc bạn học giỏi, k cho mình nhé!!!

  ta có :            \(\frac{a^3-4a^2-a+4}{a^3-7a^2+14a-8}\)

          =          \(\frac{\left(a-1\right)\left(a+1\right)\left(a-4\right)}{\left(a-4\right)\left(a-2\right)\left(a-1\right)}\)

          =           \(\frac{a+1}{a-2}\)

          nhớ nha

\(P=\frac{\left(a-4\right)\left(a-1\right)\left(a+1\right)}{\left(a-4\right)\left(a-2\right)\left(a-1\right)}=\frac{a+1}{a-1}=1+\frac{2}{a-1}\text{ }\left(a\ne4;2;1\right)\)

P nguyên khi \(\frac{2}{a-1}\) nguyên 

\(\Rightarrow a-1\in\text{Ư}\left(2\right)=\left\{-2;2;1;-1\right\}\)

\(\Rightarrow a\in\left\{-1;3;2;0\right\}\)

\(\Rightarrow a\in\left\{-1;0;3\right\}\text{ }\left(\text{do }a\ne2\right)\)

Bạn ơi 
Mình hoàn toàn đồng ý từ đầu bài nhưng đến phần bạn rút gọn là \(\frac{a+1}{a-1}\)mình thấy sai sai 
Đáng nhẽ là \(\frac{a+1}{a-2}\)chứ bạn 

\(D=\dfrac{a^3-4a^2-a+4}{a^3-7a^2+14a-8}\)

\(=\dfrac{a^2\left(a-4\right)-\left(a-4\right)}{a^3-4a^2-3a^2+12a+2a-8}\)

\(=\dfrac{\left(a^2-1\right)\left(a-4\right)}{a^2\left(a-4\right)-3a\left(a-4\right)+2\left(a-4\right)}\)

\(=\dfrac{\left(a+1\right)\left(a-1\right)\left(a-4\right)}{\left(a^2-3a+2\right)\left(a-4\right)}\)

\(=\dfrac{\left(a+1\right)\left(a-1\right)}{a^2-2a-a+2}\)

\(=\dfrac{\left(a+1\right)\left(a-1\right)}{a\left(a-2\right)-\left(a-2\right)}\)

\(=\dfrac{\left(a+1\right)\left(a-1\right)}{\left(a-1\right)\left(a-2\right)}=\dfrac{a+1}{a-2}\)

Vậy...

thay c bằng a nhé mọi người em gõ sai ạ,

ĐKXĐ: \(a\ne\pm1;2;4\)

\(P=\frac{a^3-5a^2+4a+a^2-5a+4}{a^3-5a^2+4a-2a^2+10a-8}=\frac{a\left(a^2-5a+4\right)+\left(a^2-5a+4\right)}{a\left(a^2-5a+4\right)-2\left(a^2-5a+4\right)}\)

\(P=\frac{\left(a+1\right)\left(a^2-5a+4\right)}{\left(a-2\right)\left(a^2-5a+4\right)}=\frac{a+1}{a-2}\)

b/ \(P=\frac{a+1}{a-2}=1+\frac{3}{a-2}\)

\(P\) nguyên khi \(a-2=Ư\left(3\right)=\left\{-3;-1;1;3\right\}\)

\(a-2=-3\Rightarrow a=-1\left(l\right)\)

\(a-2=-1\Rightarrow a=1\left(l\right)\)

\(a-2=1\Rightarrow a=3\)

\(a-2=3\Rightarrow a=5\)

Vậy \(\left[{}\begin{matrix}a=3\\a=5\end{matrix}\right.\) thì P nguyên

\(P=\frac{a^3-4a^2-a+4}{a^3-7a^2+14a-8}=\frac{\left(a-4\right)\left(a+1\right)\left(a-1\right)}{\left(a-1\right)\left(a-2\right)\left(a-4\right)}=\frac{a+1}{a-2}\)

b \(P=\frac{a-2+3}{a-2}=1+\frac{3}{a-2}\)

Để P nhận giá trị nguyên \(\left(a-2\right)\inƯ\left(3\right)=\left\{1;-1;-3;3\right\}\)

\(\Leftrightarrow\left[{}\begin{matrix}a-2=1\\a-2=-1\\a-2=3\\a-2=-3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}a=3\\a=1\\a=5\\a=-1\end{matrix}\right.\)

a: \(VT=\dfrac{a^2\left(a-4\right)-\left(a-4\right)}{\left(a-2\right)\left(a^2+2a+4\right)-7a\left(a-2\right)}\)

\(=\dfrac{\left(a-4\right)\left(a-1\right)\left(a+1\right)}{\left(a-1\right)\left(a^2-5a+4\right)}\)

\(=\dfrac{\left(a-4\right)\left(a+1\right)}{\left(a-4\right)\left(a-1\right)}=\dfrac{a+1}{a-1}=VP\)

b: \(VT=\dfrac{x^3\left(x+1\right)+\left(x+1\right)}{x^4-x^3+x^2+x^2-x+1}\)

\(=\dfrac{\left(x+1\right)\left(x+1\right)\left(x^2-x+1\right)}{\left(x^2+1\right)\left(x^2-x+1\right)}\)

\(=\dfrac{\left(x+1\right)^2}{x^2+1}=VP\)