Solve f(x)/g(x)=frac{[(3x^{3}+4x^{4})(-2x^{-3}-4x^{3})]-[(x^{-2}+x^{4}-6)(9x^2+6x^3)]}{(3x^3+4x^4)}

$$\frac { f ( x ) } { g ( x ) } = \frac { [ ( 3 x ^ { 3 } + 4 x ^ { 4 } ) ( - 2 x ^ { - 3 } - 4 x ^ { 3 } ) ] - [ ( x ^ { - 2 } + x ^ { 4 } - 6 ) ( 9 x ^ { 2 } + 6 x ^ { 3 } ) ] } { ( 3 x ^ { 3 } + 4 x ^ { 4 } ) }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\frac{f\left(4x+3\right)x^{3}}{22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15}\text{, }&x\neq 0\text{ and }x\neq -\frac{3}{4}\text{ and }f\neq 0\text{ and }22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15\neq 0\\g\neq 0\text{, }&x\neq 0\text{ and }x\neq -\frac{3}{4}\text{ and }22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15=0\text{ and }f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable g cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by g\left(4x+3\right)x^{3}, the least common multiple of gx,3x^{3}+4x^{4}.

$$x^{2}\left(4x+3\right)fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply x^{2} by 4x+3.

$$\left(4x^{3}+3x^{2}\right)fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}+3x^{2} by f.

$$\left(4x^{3}f+3x^{2}f\right)x=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}f+3x^{2}f by x.

$$4fx^{4}+3fx^{3}=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 3x^{3}+4x^{4} by -2x^{-3}-4x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply x^{-2}+x^{4}-6 by 9x^{2}+6x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{-2}x^{3}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{1}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

Calculate x to the power of 1 and get x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To find the opposite of 9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}, find the opposite of each term.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-9x^{-2}x^{2}-6x-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -8x and -6x to get -14x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-14x-16x^{7}-9x^{-2}x^{2}-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -12x^{6} and -9x^{6} to get -21x^{6}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-16x^{7}-9x^{-2}x^{2}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -16x^{7} and -6x^{7} to get -22x^{7}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}\right)$$

Use the distributive property to multiply g by -6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}.

$$4fx^{4}+3fx^{3}=-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}$$

Swap sides so that all variable terms are on the left hand side.

$$-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}=4fx^{4}+3fx^{3}$$

Combine all terms containing g.

$$\left(-6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}\right)g=4fx^{4}+3fx^{3}$$

The equation is in standard form.

$$\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)g=4fx^{4}+3fx^{3}$$

Divide both sides by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3}.

$$\frac{\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)g}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}$$

Dividing by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3} undoes the multiplication by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3}.

$$g=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}$$

Variable g cannot be equal to 0.

$$g=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}\text{, }g\neq 0$$

Solve for f

$$f=-\frac{g\left(22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15\right)}{\left(4x+3\right)x^{3}}$$

$$g\neq 0\text{ and }x\neq 0\text{ and }x\neq -\frac{3}{4}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by g\left(4x+3\right)x^{3}, the least common multiple of gx,3x^{3}+4x^{4}.

$$\left(4x+3\right)x^{2}fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x+3 by x^{2}.

$$\left(4x^{3}+3x^{2}\right)fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}+3x^{2} by f.

$$\left(4x^{3}f+3x^{2}f\right)x=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}f+3x^{2}f by x.

$$4fx^{4}+3fx^{3}=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 3x^{3}+4x^{4} by -2x^{-3}-4x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply x^{-2}+x^{4}-6 by 9x^{2}+6x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{-2}x^{3}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{1}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

Calculate x to the power of 1 and get x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To find the opposite of 9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}, find the opposite of each term.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-9x^{-2}x^{2}-6x-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -8x and -6x to get -14x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-14x-16x^{7}-9x^{-2}x^{2}-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -12x^{6} and -9x^{6} to get -21x^{6}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-16x^{7}-9x^{-2}x^{2}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -16x^{7} and -6x^{7} to get -22x^{7}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}\right)$$

Use the distributive property to multiply g by -6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}.

$$4fx^{4}+3fx^{3}=-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}$$

Combine all terms containing f.

$$\left(4x^{4}+3x^{3}\right)f=-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}$$

The equation is in standard form.

$$\left(4x^{4}+3x^{3}\right)f=-22gx^{7}-21gx^{6}+36gx^{3}+54gx^{2}-14gx-15g$$

Divide both sides by 3x^{3}+4x^{4}.

$$\frac{\left(4x^{4}+3x^{3}\right)f}{4x^{4}+3x^{3}}=\frac{g\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)}{4x^{4}+3x^{3}}$$

Dividing by 3x^{3}+4x^{4} undoes the multiplication by 3x^{3}+4x^{4}.

$$f=\frac{g\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)}{4x^{4}+3x^{3}}$$

Divide g\left(-22x^{7}-15-14x-21x^{6}+54x^{2}+36x^{3}\right) by 3x^{3}+4x^{4}.

$$f=\frac{g\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)}{\left(4x+3\right)x^{3}}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{f\left(4x+3\right)x^{3}}{22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15}\text{, }&x\neq -\frac{3}{4}\text{ and }x\neq 0\text{ and }f\neq 0\text{ and }22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15\neq 0\\g\neq 0\text{, }&x\neq 0\text{ and }x\neq -\frac{3}{4}\text{ and }22x^{7}+21x^{6}-36x^{3}-54x^{2}+14x+15=0\text{ and }f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable g cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by g\left(4x+3\right)x^{3}, the least common multiple of gx,3x^{3}+4x^{4}.

$$x^{2}\left(4x+3\right)fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply x^{2} by 4x+3.

$$\left(4x^{3}+3x^{2}\right)fx=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}+3x^{2} by f.

$$\left(4x^{3}f+3x^{2}f\right)x=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 4x^{3}f+3x^{2}f by x.

$$4fx^{4}+3fx^{3}=g\left(\left(3x^{3}+4x^{4}\right)\left(-2x^{-3}-4x^{3}\right)-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply 3x^{3}+4x^{4} by -2x^{-3}-4x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(x^{-2}+x^{4}-6\right)\left(9x^{2}+6x^{3}\right)\right)$$

Use the distributive property to multiply x^{-2}+x^{4}-6 by 9x^{2}+6x^{3}.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{-2}x^{3}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x^{1}+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

Calculate x to the power of 1 and get x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-\left(9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}\right)\right)$$

To find the opposite of 9x^{-2}x^{2}+6x+9x^{6}+6x^{7}-54x^{2}-36x^{3}, find the opposite of each term.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-8x-16x^{7}-9x^{-2}x^{2}-6x-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -8x and -6x to get -14x.

$$4fx^{4}+3fx^{3}=g\left(-6-12x^{6}-14x-16x^{7}-9x^{-2}x^{2}-9x^{6}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -12x^{6} and -9x^{6} to get -21x^{6}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-16x^{7}-9x^{-2}x^{2}-6x^{7}+54x^{2}+36x^{3}\right)$$

Combine -16x^{7} and -6x^{7} to get -22x^{7}.

$$4fx^{4}+3fx^{3}=g\left(-6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}\right)$$

Use the distributive property to multiply g by -6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}.

$$4fx^{4}+3fx^{3}=-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}$$

Swap sides so that all variable terms are on the left hand side.

$$-6g-21gx^{6}-14gx-22gx^{7}-9gx^{-2}x^{2}+54gx^{2}+36gx^{3}=4fx^{4}+3fx^{3}$$

Combine all terms containing g.

$$\left(-6-21x^{6}-14x-22x^{7}-9x^{-2}x^{2}+54x^{2}+36x^{3}\right)g=4fx^{4}+3fx^{3}$$

The equation is in standard form.

$$\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)g=4fx^{4}+3fx^{3}$$

Divide both sides by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3}.

$$\frac{\left(-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15\right)g}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}$$

Dividing by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3} undoes the multiplication by -15-14x-21x^{6}-22x^{7}+54x^{2}+36x^{3}.

$$g=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}$$

Variable g cannot be equal to 0.

$$g=\frac{f\left(4x+3\right)x^{3}}{-22x^{7}-21x^{6}+36x^{3}+54x^{2}-14x-15}\text{, }g\neq 0$$