Solve 3x^{2}+5x+7+2x^3+3x+16/x^2+x=10x^3+12x+4/x-2+7x^3/x+1

$$3 x ^ { 2 } + 5 x + 7 + \frac { 2 x ^ { 3 } + 3 x + 16 } { x ^ { 2 } + x } = \frac { 10 x ^ { 3 } + 12 x + 4 } { x } - \frac { 2 + 7 x ^ { 3 } } { x + 1 }$$

Solve for x

$$x=3$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x^{2}+x,x,x+1.

$$3x^{2}x\left(x+1\right)+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

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

$$3x^{3}\left(x+1\right)+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply 3x^{3} by x+1.

$$3x^{4}+3x^{3}+5xx\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Multiply x and x to get x^{2}.

$$3x^{4}+3x^{3}+5x^{2}\left(x+1\right)+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply 5x^{2} by x+1.

$$3x^{4}+3x^{3}+5x^{3}+5x^{2}+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Combine 3x^{3} and 5x^{3} to get 8x^{3}.

$$3x^{4}+8x^{3}+5x^{2}+x\left(x+1\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply x by x+1.

$$3x^{4}+8x^{3}+5x^{2}+\left(x^{2}+x\right)\times 7+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply x^{2}+x by 7.

$$3x^{4}+8x^{3}+5x^{2}+7x^{2}+7x+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Combine 5x^{2} and 7x^{2} to get 12x^{2}.

$$3x^{4}+8x^{3}+12x^{2}+7x+2x^{3}+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Combine 8x^{3} and 2x^{3} to get 10x^{3}.

$$3x^{4}+10x^{3}+12x^{2}+7x+3x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Combine 7x and 3x to get 10x.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=\left(x+1\right)\left(10x^{3}+12x+4\right)-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply x+1 by 10x^{3}+12x+4 and combine like terms.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-x\left(2+7x^{3}\right)$$

Use the distributive property to multiply x by 2+7x^{3}.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-\left(2x+7x^{4}\right)$$

To find the opposite of 2x+7x^{4}, find the opposite of each term.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+16x+10x^{3}+4-2x-7x^{4}$$

Combine 16x and -2x to get 14x.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=10x^{4}+12x^{2}+14x+10x^{3}+4-7x^{4}$$

Combine 10x^{4} and -7x^{4} to get 3x^{4}.

$$3x^{4}+10x^{3}+12x^{2}+10x+16=3x^{4}+12x^{2}+14x+10x^{3}+4$$

Subtract 3x^{4} from both sides.

$$3x^{4}+10x^{3}+12x^{2}+10x+16-3x^{4}=12x^{2}+14x+10x^{3}+4$$

Combine 3x^{4} and -3x^{4} to get 0.

$$10x^{3}+12x^{2}+10x+16=12x^{2}+14x+10x^{3}+4$$

Subtract 12x^{2} from both sides.

$$10x^{3}+12x^{2}+10x+16-12x^{2}=14x+10x^{3}+4$$

Combine 12x^{2} and -12x^{2} to get 0.

$$10x^{3}+10x+16=14x+10x^{3}+4$$

Subtract 14x from both sides.

$$10x^{3}+10x+16-14x=10x^{3}+4$$

Combine 10x and -14x to get -4x.

$$10x^{3}-4x+16=10x^{3}+4$$

Subtract 10x^{3} from both sides.

$$10x^{3}-4x+16-10x^{3}=4$$

Combine 10x^{3} and -10x^{3} to get 0.

$$-4x+16=4$$

Subtract 16 from both sides.

$$-4x=4-16$$

Subtract 16 from 4 to get -12.

$$-4x=-12$$

Divide both sides by -4.

$$x=\frac{-12}{-4}$$

Divide -12 by -4 to get 3.

$$x=3$$