Rút gọn các phân thức sau:

a) \(\frac{{9 - 12x + 4{x^2}}}{{2x - 3}}\);

b) \(\frac{{{{\left( {2x + 3} \right)}^2} + 2\left( {4{x^2} - 9} \right) + {{\left( {2x - 3} \right)}^2}}}{{{{\left( {2x - 3} \right)}^2} - 2\left( {4{x^2} - 9} \right) + {{\left( {2x + 3} \right)}^2}}}\);

c) \(\frac{{{{\left( {2x + 3} \right)}^3} - {{\left( {2x - 3} \right)}^3}}}{{{{\left( {3x + 4} \right)}^2} + 3{x^2} - 24x - 7}}\);

d) \(\frac{{\left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) + 1}}{{{x^2} + 5x + 5}}\);

e) \(\frac{{{x^4} + 4}}{{x\left( {{x^2} + 2} \right) - 2{x^2} - {{\left( {x - 2} \right)}^2} - 1}}\).

Lời giải

a) \(\frac{{9 - 12x + 4{x^2}}}{{2x - 3}} = \frac{{{{\left( {2x - 3} \right)}^2}}}{{2x - 3}} = 2x - 3\).

b) \(\frac{{{{\left( {2x + 3} \right)}^2} + 2\left( {4{x^2} - 9} \right) + {{\left( {2x - 3} \right)}^2}}}{{{{\left( {2x - 3} \right)}^2} - 2\left( {4{x^2} - 9} \right) + {{\left( {2x + 3} \right)}^2}}}\)

\( = \frac{{{{\left( {2x + 3} \right)}^2} + 2\left( {2x + 3} \right)\left( {2x - 3} \right) + {{\left( {2x - 3} \right)}^2}}}{{{{\left( {2x - 3} \right)}^2} - 2\left( {2x + 3} \right)\left( {2x - 3} \right) + {{\left( {2x + 3} \right)}^2}}}\)

\( = \frac{{{{\left( {2x + 3 + 2x - 3} \right)}^2}}}{{{{\left( {2x - 3 - 2x - 3} \right)}^2}}}\)

\( = \frac{{{{\left( {4x} \right)}^2}}}{{{{\left( { - 6} \right)}^2}}} = \frac{{16{x^2}}}{{36}} = \frac{{4{x^2}}}{9}\).

c) \(\frac{{{{\left( {2x + 3} \right)}^3} - {{\left( {2x - 3} \right)}^3}}}{{{{\left( {3x + 4} \right)}^2} + 3{x^2} - 24x - 7}}\)

\( = \frac{{\left( {2x + 3 - 2x + 3} \right)\left[ {{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right)\left( {2x - 3} \right) + {{\left( {2x - 3} \right)}^2}} \right]}}{{9{x^2} + 24x + 16 + 3{x^2} - 24x - 7}}\)

\( = \frac{{6\left( {4{x^2} + 12x + 9 + 4{x^2} - 9 + 4{x^2} - 12x + 9} \right)}}{{12{x^2} + 9}}\)

\( = \frac{{6\left( {12{x^2} + 9} \right)}}{{12{x^2} + 9}} = 6\).

d) \(\frac{{\left( {x + 1} \right)\left( {x + 2} \right)\left( {x + 3} \right)\left( {x + 4} \right) + 1}}{{{x^2} + 5x + 5}}\)

\( = \frac{{\left( {{x^2} + 5x + 4} \right)\left( {{x^2} + 5x + 6} \right) + 1}}{{{x^2} + 5x + 5}}\)

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

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

\( = \frac{{{{\left( {{x^2} + 5x + 5} \right)}^2}}}{{{x^2} + 5x + 5}} = {x^2} + 5x + 5\).

e) \(\frac{{{x^4} + 4}}{{x\left( {{x^2} + 2} \right) - 2{x^2} - {{\left( {x - 2} \right)}^2} - 1}}\)

\( = \frac{{{x^4} + 4}}{{{x^3} + 2x - 2{x^2} - \left( {{x^2} - 4x + 4} \right) - 1}}\)

\( = \frac{{{x^4} + 4}}{{{x^3} - 3{x^2} + 6x - 5}}\).