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\frac{1}{6}x-\left(2x\times \frac{1}{2}x-2x+\frac{1}{2}x-1-\left(3x+1\right)\left(\frac{1}{3}x-1\right)\right)+x
Apply the distributive property by multiplying each term of 2x+1 by each term of \frac{1}{2}x-1.
\frac{1}{6}x-\left(2x^{2}\times \frac{1}{2}-2x+\frac{1}{2}x-1-\left(3x+1\right)\left(\frac{1}{3}x-1\right)\right)+x
Multiply x and x to get x^{2}.
\frac{1}{6}x-\left(x^{2}-2x+\frac{1}{2}x-1-\left(3x+1\right)\left(\frac{1}{3}x-1\right)\right)+x
Cancel out 2 and 2.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-\left(3x+1\right)\left(\frac{1}{3}x-1\right)\right)+x
Combine -2x and \frac{1}{2}x to get -\frac{3}{2}x.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-\left(3x\times \frac{1}{3}x-3x+\frac{1}{3}x-1\right)\right)+x
Apply the distributive property by multiplying each term of 3x+1 by each term of \frac{1}{3}x-1.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-\left(3x^{2}\times \frac{1}{3}-3x+\frac{1}{3}x-1\right)\right)+x
Multiply x and x to get x^{2}.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-\left(x^{2}-3x+\frac{1}{3}x-1\right)\right)+x
Cancel out 3 and 3.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-\left(x^{2}-\frac{8}{3}x-1\right)\right)+x
Combine -3x and \frac{1}{3}x to get -\frac{8}{3}x.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-x^{2}-\left(-\frac{8}{3}x\right)-\left(-1\right)\right)+x
To find the opposite of x^{2}-\frac{8}{3}x-1, find the opposite of each term.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-x^{2}+\frac{8}{3}x-\left(-1\right)\right)+x
The opposite of -\frac{8}{3}x is \frac{8}{3}x.
\frac{1}{6}x-\left(x^{2}-\frac{3}{2}x-1-x^{2}+\frac{8}{3}x+1\right)+x
The opposite of -1 is 1.
\frac{1}{6}x-\left(-\frac{3}{2}x-1+\frac{8}{3}x+1\right)+x
Combine x^{2} and -x^{2} to get 0.
\frac{1}{6}x-\left(\frac{7}{6}x-1+1\right)+x
Combine -\frac{3}{2}x and \frac{8}{3}x to get \frac{7}{6}x.
\frac{1}{6}x-\frac{7}{6}x+x
Add -1 and 1 to get 0.
-x+x
Combine \frac{1}{6}x and -\frac{7}{6}x to get -x.
0
Combine -x and x to get 0.
\frac{x-\left(3\left(2x+1\right)\left(x-2\right)-2\left(3x+1\right)\left(x-3\right)\right)+6x}{6}
Factor out \frac{1}{6}.
0
Simplify.
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