Rút gọn biểu thức \(P = \dfrac{{2\sqrt a }}{{\sqrt a  + 3}} + \dfrac{{\sqrt a  + 1}}{{\sqrt a  - 3}} + \dfrac{{3 + 7\sqrt a }}{{9 - a}}\) với \(a \ge 0,\,\,a \ne 9\).

Với \(a \ge 0,\,\,a \ne 9\) ta có:

\(\begin{array}{l}P = \dfrac{{2\sqrt a }}{{\sqrt a  + 3}} + \dfrac{{\sqrt a  + 1}}{{\sqrt a  - 3}} + \dfrac{{3 + 7\sqrt a }}{{9 - a}}\\P = \dfrac{{2\sqrt a }}{{\sqrt a  + 3}} + \dfrac{{\sqrt a  + 1}}{{\sqrt a  - 3}} - \dfrac{{3 + 7\sqrt a }}{{\left( {\sqrt a  + 3} \right)\left( {\sqrt a  - 3} \right)}}\\P = \dfrac{{2\sqrt a \left( {\sqrt a  - 3} \right) + \left( {\sqrt a  + 1} \right)\left( {\sqrt a  + 3} \right) - \left( {3 + 7\sqrt a } \right)}}{{\left( {\sqrt a  + 3} \right)\left( {\sqrt a  - 3} \right)}}\end{array}\)

\(\begin{array}{l}P = \dfrac{{2a - 6\sqrt a  + a + 3\sqrt a  + \sqrt a  + 3 - 3 - 7\sqrt a }}{{\left( {\sqrt a  + 3} \right)\left( {\sqrt a  - 3} \right)}}\\P = \dfrac{{3a - 9\sqrt a }}{{\left( {\sqrt a  + 3} \right)\left( {\sqrt a  - 3} \right)}}\\P = \dfrac{{3\sqrt a \left( {\sqrt a  - 3} \right)}}{{\left( {\sqrt a  + 3} \right)\left( {\sqrt a  - 3} \right)}}\\P = \dfrac{{3\sqrt a }}{{\sqrt a  + 3}}\end{array}\)

Vậy với \(a \ge 0,\,\,a \ne 9\) thì \(P = \dfrac{{3\sqrt a }}{{\sqrt a  + 3}}\).