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gx\times 4x-fx\times 4x=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply both sides of the equation by 4x, the least common multiple of 4,4x.
gx^{2}\times 4-fx\times 4x=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-fx^{2}\times 4=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-4fx^{2}=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply -1 and 4 to get -4.
gx^{2}\times 4-4fx^{2}=4x^{3}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+x-\left(5x+4\right)
Multiply 4 and \frac{1}{4} to get 1.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+x-5x-4
To find the opposite of 5x+4, find the opposite of each term.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}-4x-4
Combine x and -5x to get -4x.
-4fx^{2}=4x^{3}+4x^{2}-4x-4-gx^{2}\times 4
Subtract gx^{2}\times 4 from both sides.
-4fx^{2}=4x^{3}+4x^{2}-4x-4-4gx^{2}
Multiply -1 and 4 to get -4.
\left(-4x^{2}\right)f=4x^{3}-4gx^{2}+4x^{2}-4x-4
The equation is in standard form.
\frac{\left(-4x^{2}\right)f}{-4x^{2}}=\frac{4x^{3}-4gx^{2}+4x^{2}-4x-4}{-4x^{2}}
Divide both sides by -4x^{2}.
f=\frac{4x^{3}-4gx^{2}+4x^{2}-4x-4}{-4x^{2}}
Dividing by -4x^{2} undoes the multiplication by -4x^{2}.
f=g-x-1+\frac{1}{x}+\frac{1}{x^{2}}
Divide 4x^{3}+4x^{2}-4x-4-4gx^{2} by -4x^{2}.
gx\times 4x-fx\times 4x=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply both sides of the equation by 4x, the least common multiple of 4,4x.
gx^{2}\times 4-fx\times 4x=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-fx^{2}\times 4=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-4fx^{2}=4xx^{2}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply -1 and 4 to get -4.
gx^{2}\times 4-4fx^{2}=4x^{3}+4xx+4x\times \frac{1}{4}-\left(5x+4\right)
To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+4x\times \frac{1}{4}-\left(5x+4\right)
Multiply x and x to get x^{2}.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+x-\left(5x+4\right)
Multiply 4 and \frac{1}{4} to get 1.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}+x-5x-4
To find the opposite of 5x+4, find the opposite of each term.
gx^{2}\times 4-4fx^{2}=4x^{3}+4x^{2}-4x-4
Combine x and -5x to get -4x.
gx^{2}\times 4=4x^{3}+4x^{2}-4x-4+4fx^{2}
Add 4fx^{2} to both sides.
4x^{2}g=4x^{3}+4fx^{2}+4x^{2}-4x-4
The equation is in standard form.
\frac{4x^{2}g}{4x^{2}}=\frac{4x^{3}+4fx^{2}+4x^{2}-4x-4}{4x^{2}}
Divide both sides by 4x^{2}.
g=\frac{4x^{3}+4fx^{2}+4x^{2}-4x-4}{4x^{2}}
Dividing by 4x^{2} undoes the multiplication by 4x^{2}.
g=f+x+1-\frac{x+1}{x^{2}}
Divide 4x^{3}+4x^{2}-4x-4+4fx^{2} by 4x^{2}.