Solve for x:
$fraction numerator 1 over denominator straight x plus 1 end fraction plus fraction numerator 2 over denominator x plus 2 end fraction plus fraction numerator 4 over denominator x plus 4 end fraction comma space x not equal to negative 1 comma negative 2 comma negative 4$

fraction numerator 1 over denominator straight x plus 1 end fraction plus fraction numerator 2 over denominator straight x plus 2 end fraction plus fraction numerator 4 over denominator straight x plus 4 end fraction straight L. straight C. straight M space of space all space the space denominators space is space left parenthesis straight x plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis left parenthesis straight x plus 4 right parenthesis Multiply space throughtoout space by space the space straight L. straight C. straight M space we space get left parenthesis straight x plus 2 right parenthesis left parenthesis straight x plus 4 right parenthesis plus 2 left parenthesis straight x plus 1 right parenthesis left parenthesis straight x plus 4 right parenthesis space equals 4 left parenthesis straight x plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis therefore left parenthesis straight x plus 4 right parenthesis left parenthesis straight x plus 2 plus 2 straight x plus 2 right parenthesis equals 4 left parenthesis straight x squared plus 3 straight x plus 2 right parenthesis therefore space left parenthesis straight x plus 4 right parenthesis left parenthesis 3 straight x plus 4 right parenthesis space equals 4 straight x squared plus 12 straight x space plus 8 therefore space 3 straight x squared plus 16 straight x plus 16 space equals space 4 straight x squared plus 12 straight x plus 8 therefore space straight x squared minus 4 straight x minus 8 equals 0 Now space straight a comma space equals 1 space straight b equals negative 4 comma space straight c equals negative 8 straight x equals fraction numerator negative straight b begin display style plus for minus of end style square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction equals fraction numerator 4 plus for minus of square root of 16 plus 32 end root over denominator 2 end fraction space equals fraction numerator 4 plus for minus of square root of 48 over denominator 2 end fraction space equals fraction numerator 4 plus for minus of 4 square root of 3 over denominator 2 end fraction space straight x equals 2 plus for minus of 2 square root of 3

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