ĐKXĐ: ....
\(P=\frac{\sqrt{a}\left(a\sqrt{a}-1\right)}{a+\sqrt{a}+1}-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}}+\frac{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}{\sqrt{a}+2}\)
\(=\frac{\sqrt{a}\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}{a+\sqrt{a}+1}-\left(\sqrt{a}-1\right)+\sqrt{a}-2\)
\(=a-\sqrt{a}-\sqrt{a}+1+\sqrt{a}-2\)
\(=a-\sqrt{a}-1\)
\(P=0\Rightarrow a-\sqrt{a}-1=0\Rightarrow\left[{}\begin{matrix}\sqrt{a}=\frac{1+\sqrt{5}}{2}\\\sqrt{a}=\frac{1-\sqrt{5}}{2}< 0\left(l\right)\end{matrix}\right.\)
\(\Rightarrow a=\left(\frac{1+\sqrt{5}}{2}\right)^2=\frac{3+\sqrt{5}}{2}\)
c/ Đặt \(x=2003\)
\(a=\frac{\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}{\left[\left(x+5\right)\left(x+9\right)-\left(x+3\right)\right]\left[x\left(x+5\right)+12\left(x+6\right)\right]}=\frac{\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}{\left(x^2+14x+45-x-3\right)\left(x^2+5x+12x+72\right)}\)
\(=\frac{\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}{\left(x^2+13x+42\right)\left(x^2+17x+72\right)}=\frac{\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}{\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}=1\)
\(\Rightarrow a=1\Rightarrow P=-1\)