Đáp án:
\(\dfrac{{4{x^4} + {x^3} - 34{x^2} + 12x - 32}}{{4{x^2}{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}\)
Giải thích các bước giải:
\(\begin{array}{l}
DK:x \ne \pm 2;x \ne 0\\
\dfrac{1}{{x + 2}} - \dfrac{3}{{4{x^2}}} + \dfrac{{x - 14}}{{\left( {{x^2} + 4x + 4} \right)\left( {x - 2} \right)}}\\
= \dfrac{1}{{x + 2}} - \dfrac{3}{{4{x^2}}} + \dfrac{{x - 14}}{{{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}\\
= \dfrac{{4{x^2}\left( {x + 2} \right)\left( {x - 2} \right) - 3\left( {{x^2} + 4x + 4} \right)\left( {x - 2} \right) + 4{x^2}\left( {x - 14} \right)}}{{4{x^2}{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}\\
= \dfrac{{4{x^2}\left( {{x^2} - 4} \right) - \left( {3{x^2} + 12x + 12} \right)\left( {x - 2} \right) + 4{x^3} - 56}}{{4{x^2}{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}\\
= \dfrac{{4{x^4} - 16{x^2} - 3{x^3} - 6{x^2} - 12{x^2} + 24x - 12x + 24 + 4{x^3} - 56}}{{4{x^2}{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}\\
= \dfrac{{4{x^4} + {x^3} - 34{x^2} + 12x - 32}}{{4{x^2}{{\left( {x + 2} \right)}^2}\left( {x - 2} \right)}}
\end{array}\)
