Question 73765
{{{5 / x + 2  +  2 / x ^ 2  +  7 x + 6  =  3 / x + 1}}}
{{{x^2(5 / x + 2  +  2 / x ^ 2  +  7 x + 6)  =  x^2(3 / x + 1)}}}To get rid of the fractions, multiply everything by the LCD (the LCD is x^2)
{{{5x + 2x^2  +  2  +  7x^3 + 6x^2  =  3x + x^2}}}Distribute the x^2
{{{7x^3 + 8x^2 + 5x   +  2 =  3x + x^2}}}Combine like terms 
{{{7x^3 + 8x^2 -x^2 + 5x -3x  +  2 = 0}}}Get everything to one side
{{{highlight(7x^3 + 7x^2) + highlight(2x  +  2) = 0}}}Group the highlighted terms
{{{7x^2(x + 1) + 2(x  +  1) = 0}}}Factor out a 7x^2 out of the 1st group of parenthesis and factor out a 2 out of the 2nd group of parenthesis. Notice how we have a common term of (x+1). We can combine them now
{{{(7x^2+2)(x + 1)= 0}}}Combine the terms. If we let y=(x+1) it would look like {{{7x^2y+2y=(7x^2+2)(y)}}} Now we can solve for x.
{{{(7x^2+2)(x + 1)= 0}}} Isolate each factor and solve:
{{{(x + 1)= 0}}}
{{{x=-1}}}There's our answer
{{{(7x^2+2)=0}}}
{{{7x^2=-2}}}
{{{x^2=-2/7}}}
{{{x=sqrt(-2/7)}}}This answer will not be real (I don't know if you know about complex roots yet), so I would ignore this factor.
So our answer is x=-1
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Check
{{{5 / x + 2  +  2 / x ^ 2  +  7 x + 6  =  3 / x + 1}}}Plug in x=-1
{{{5 /(-1) + 2  +  2 /(-1)^ 2  +  7(-1) + 6  =  3 /(-1)+ 1}}}
{{{-5 + 2  +  2  -7 + 6  =  -3+ 1}}}
{{{-3  +  -5 + 6  =  -2}}}
{{{-2  =  -2}}}Works