Question 194424
You made a mistake in assuming that {{{-x^2-x=-x^3}}} which is NOT true. You CANNOT combine variables of different powers.



What you need to do is solve {{{12 - x^2 - x = 0}}}



{{{12 - x^2 - x = 0}}} Start with the given equation.



{{{- x^2 - x + 12= 0}}} Rearrange the terms.



{{{x^2 + x - 12= 0}}} Multiply EVERY term by -1 to make the leading coefficient positive.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=1}}}, and {{{c=-12}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(1) +- sqrt( (1)^2-4(1)(-12) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=1}}}, and {{{c=-12}}}



{{{x = (-1 +- sqrt( 1-4(1)(-12) ))/(2(1))}}} Square {{{1}}} to get {{{1}}}. 



{{{x = (-1 +- sqrt( 1--48 ))/(2(1))}}} Multiply {{{4(1)(-12)}}} to get {{{-48}}}



{{{x = (-1 +- sqrt( 1+48 ))/(2(1))}}} Rewrite {{{sqrt(1--48)}}} as {{{sqrt(1+48)}}}



{{{x = (-1 +- sqrt( 49 ))/(2(1))}}} Add {{{1}}} to {{{48}}} to get {{{49}}}



{{{x = (-1 +- sqrt( 49 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-1 +- 7)/(2)}}} Take the square root of {{{49}}} to get {{{7}}}. 



{{{x = (-1 + 7)/(2)}}} or {{{x = (-1 - 7)/(2)}}} Break up the expression. 



{{{x = (6)/(2)}}} or {{{x =  (-8)/(2)}}} Combine like terms. 



{{{x = 3}}} or {{{x = -4}}} Simplify. 



So the solutions are {{{x = 3}}} or {{{x = -4}}}




To verify the solutions, simply plug them back into {{{x/3 = 4/(x+1)}}}