Question 324935
{{{(a+b)(a-b)(1+x)(1-x)=4abx}}}
{{{(1+x)(1-x)=(4ab)/((a+b)(a-b))x}}}
{{{1-x^2=((4ab)/((a^2-b^2)))x}}}
{{{x^2+ ((4ab)/((a^2-b^2))) x-1=0}}}
Use the quadratic formula, careful not to confuse the coefficients (a,b,c),
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
{{{x = (-((4ab)/((a^2-b^2))) +- sqrt( ((4ab)/((a^2-b^2)))^2-4*1*(-1) ))/2 }}}
Let's look at the term under the square root sign separately,
 {{{((4ab)/((a^2-b^2)))^2+4=((4ab)/((a^2-b^2)))^2+(4(a^2-b^2)^2)/(a^2-b^2)^2}}}
{{{((4ab)/((a^2-b^2)))^2+4=((4ab)^2+4(a^2-b^2)^2)/(a^2-b^2)^2}}}
{{{((4ab)/((a^2-b^2)))^2+4=(16a^2b^2+(4a^4-8a^2b^2+4b^4))/(a^2-b^2)^2}}}
{{{((4ab)/((a^2-b^2)))^2+4=(4a^4+8a^2b^2+4b^4)/(a^2-b^2)^2}}}
{{{((4ab)/((a^2-b^2)))^2+4=(4(a^2+b^2)^2)/(a^2-b^2)^2}}}
Now back to the square root.
{{{x = (-((4ab)/((a^2-b^2))) +- sqrt((4(a^2+b^2)^2)/(a^2-b^2)^2  ))/2 }}}
{{{x = (-((4ab)/((a^2-b^2))) +- (2(a^2+b^2))/(a^2-b^2)  )/2 }}}
Two solutions,
{{{x = (-((4ab)/((a^2-b^2))) + (2(a^2+b^2))/(a^2-b^2)  )/2 }}}
{{{x = (a^2-2ab+b^2)/(a^2-b^2) }}}
{{{x=(a-b)^2/((a+b)(a-b))}}}
{{{highlight(x=(a-b)/(a+b))}}}
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{{{x = (-((4ab)/((a^2-b^2))) - (2(a^2+b^2))/(a^2-b^2)  )/2 }}}
{{{x = -(a^2+2ab+b^2)/(a^2-b^2) }}}
{{{x=-(a+b)^2/((a+b)(a-b))}}}
{{{highlight(x=-(a+b)/(a-b))}}}