Question 14329
{{{((x-3)/(x-2))+((x+1)/(x+3)) = (2x^2+x+1)/(x^2+x-6)}}} Add the fractions on the left side.
{{{((x+3)(x-3)+(x+1)(x-2))/(x-2)(x+3) = (2x^2+x+1)/(x^2+x-6)}}} On the left side, multiply the factors in the numerator and multiply the factors in the denominator.
{{{((x^2-9)+(x^2-x-2))/(x^2+x-6) = (2x^2+x+1)/(x^2+x-6)}}} Simplify.
{{{2x^2-x-11 = 2x^2+x+1}}} Subtract {{{2x^2}}} from both sides.
{{{-x-11 = x+1}}} Add x to both sides.
{{{-11 = 2x+1}}} Subtract 1 from both sides.
{{{-12 = 2x}}} Divide both sides by 2.
{{{-6 = x}}}

So, x = -6

Check:
{{{((-6-3)/(-6-2))+((-6+1)/(-6+3)) = (2(-6)^2 +(-6)+1)/((-6)^2+(-6)-6)}}}
{{{(-9/-8)+(-5/-3) = (72-6+1)/(36-6-6)}}}
{{{(27+40)/24 = 67/24}}}
{{{67/24 = 67/24}}}

x = -6 is correct, not x = 6 as you say your book shows.