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$\frac{1}{x+2}+\frac{3}{{{x}^{2}}-4}+\frac{x-14}{({{x}^{2}}+4x+4)(x-2)}$
$=\frac{1}{x+2}+\frac{3}{\left( x+2 \right)\left( x-2 \right)}+\frac{x-14}{{{(x+2)}^{2}}(x-2)}$
$=\frac{\left( x+2 \right)\left( x-2 \right)+3\left( x+2 \right)+x-14}{{{(x+2)}^{2}}(x-2)}$
$=\frac{{{x}^{2}}-4+3x+6+x-14}{{{(x+2)}^{2}}(x-2)}$
$=\frac{{{x}^{2}}+4x-12}{{{(x+2)}^{2}}(x-2)}$
$=\frac{{{x}^{2}}+6x-2x-12}{{{(x+2)}^{2}}(x-2)}$
$=\frac{(x+6)(x-2)}{{{(x+2)}^{2}}(x-2)}$
$=\frac{x+6}{{{(x+2)}^{2}}}$