$$ \left. \begin{array} { | l | } \hline ext { Perform the indicated operation(s). } \ { \left. \begin{array} { c } { ( x ^ { 2 } - x + 3 x ^ { 3 } - 7 ) + ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 ) } \ { 2 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - x + 6 } \ { 2 x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 3 x + 10 } \end{array} ight. } \ \hline \left. \begin{array} { c } { ( 2 x ^ { 3 } - 5 x ^ { 2 } + 7 ) - ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 } \ { ( 2 x ^ { 3 } + 5 x ^ { 2 } + x + 2 ) + ( 5 x ^ { 3 } + 7 x + 8 ) + ( x ^ { 2 } - x ^ { 3 } + 12 x ) - ( x ^ { 2 } - x ^ { 3 } + 12 x ) } \ { 6 x ^ { 3 } + 4 x ^ { 2 } - 6 x + 6 } \end{array} ight. \end{array} ight. $$

Question

$$ \left. \begin{array} { | l | } \hline 	ext { Perform the indicated operation(s). } \ { \left. \begin{array} { c } { ( x ^ { 2 } - x + 3 x ^ { 3 } - 7 ) + ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 ) } \ { 2 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - x + 6 } \ { 2 x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 3 x + 10 } \end{array} ight. } \ \hline \left. \begin{array} { c } { ( 2 x ^ { 3 } - 5 x ^ { 2 } + 7 ) - ( x ^ { 3 } - 5 x ^ { 2 } + 2 x ^ { 4 } - x + 6 } \ { ( 2 x ^ { 3 } + 5 x ^ { 2 } + x + 2 ) + ( 5 x ^ { 3 } + 7 x + 8 ) + ( x ^ { 2 } - x ^ { 3 } + 12 x ) - ( x ^ { 2 } - x ^ { 3 } + 12 x ) } \ { 6 x ^ { 3 } + 4 x ^ { 2 } - 6 x + 6 } \end{array} ight. \end{array} ight. $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Simplify:
  $$\left(2x^{3}-5x^{2}+7\right)-\left(x^{3}-5x^{2}+2x^{4}-x+6\right)$$
- step1: Remove the parentheses:
  $$2x^{3}-5x^{2}+7-\left(x^{3}-5x^{2}+2x^{4}-x+6\right)$$
- step2: Remove the parentheses:
  $$2x^{3}-5x^{2}+7-x^{3}+5x^{2}-2x^{4}+x-6$$
- step3: Subtract the terms:
  $$x^{3}+1-2x^{4}+x$$
Expand the expression $$ (x^{2} - x + 3x^{3} - 7) + (x^{3} - 5x^{2} + 2x^{4} - x + 6) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\left(x^{2}-x+3x^{3}-7\right)+\left(x^{3}-5x^{2}+2x^{4}-x+6\right)$$
- step1: Remove the parentheses:
  $$x^{2}-x+3x^{3}-7+x^{3}-5x^{2}+2x^{4}-x+6$$
- step2: Subtract the terms:
  $$-4x^{2}-2x+4x^{3}-1+2x^{4}$$
Expand the expression $$ (2x^{3} + 5x^{2} + x + 2) + (5x^{3} + 7x + 8) + (x^{2} - x^{3} + 12x) - (x^{2} - x^{3} + 12x) $$
Simplify the expression by following steps:
- step0: Simplify:
  $$\left(2x^{3}+5x^{2}+x+2\right)+\left(5x^{3}+7x+8\right)+\left(x^{2}-x^{3}+12x\right)-\left(x^{2}-x^{3}+12x\right)$$
- step1: Remove the parentheses:
  $$2x^{3}+5x^{2}+x+2+5x^{3}+7x+8+x^{2}-x^{3}+12x-\left(x^{2}-x^{3}+12x\right)$$
- step2: Remove the parentheses:
  $$2x^{3}+5x^{2}+x+2+5x^{3}+7x+8+x^{2}-x^{3}+12x-x^{2}+x^{3}-12x$$
- step3: Calculate:
  $$7x^{3}+5x^{2}+8x+10$$
Let's perform the indicated operations step by step.

### First Operation
We need to simplify the expression:
$$
(x^{2} - x + 3x^{3} - 7) + (x^{3} - 5x^{2} + 2x^{4} - x + 6)
$$

1. Combine like terms:
   - $$x^{4}$$: $$2x^{4}$$
   - $$x^{3}$$: $$3x^{3} + x^{3} = 4x^{3}$$
   - $$x^{2}$$: $$x^{2} - 5x^{2} = -4x^{2}$$
   - $$x$$: $$-x - x = -2x$$
   - Constant: $$-7 + 6 = -1$$

Thus, the result of the first operation is:
$$
2x^{4} + 4x^{3} - 4x^{2} - 2x - 1
$$

### Second Operation
Now, we simplify:
$$
(2x^{3} - 5x^{2} + 7) - (x^{3} - 5x^{2} + 2x^{4} - x + 6)
$$

1. Distribute the negative sign:
   - $$2x^{3} - 5x^{2} + 7 - x^{3} + 5x^{2} - 2x^{4} + x - 6$$

2. Combine like terms:
   - $$x^{4}$$: $$-2x^{4}$$
   - $$x^{3}$$: $$2x^{3} - x^{3} = x^{3}$$
   - $$x^{2}$$: $$-5x^{2} + 5x^{2} = 0$$
   - $$x$$: $$x$$
   - Constant: $$7 - 6 = 1$$

Thus, the result of the second operation is:
$$
-2x^{4} + x^{3} + x + 1
$$

### Third Operation
Finally, we simplify:
$$
(2x^{3} + 5x^{2} + x + 2) + (5x^{3} + 7x + 8) + (x^{2} - x^{3} + 12x) - (x^{2} - x^{3} + 12x)
$$

1. The last two terms cancel each other out:
   - $$(x^{2} - x^{3} + 12x) - (x^{2} - x^{3} + 12x) = 0$$

2. Combine the remaining terms:
   - $$2x^{3} + 5x^{3} = 7x^{3}$$
   - $$5x^{2}$$
   - $$x + 7x = 8x$$
   - Constant: $$2 + 8 = 10$$

Thus, the result of the third operation is:
$$
7x^{3} + 5x^{2} + 8x + 10
$$

### Summary of Results
1. First operation: $$2x^{4} + 4x^{3} - 4x^{2} - 2x - 1$$
2. Second operation: $$-2x^{4} + x^{3} + x + 1$$
3. Third operation: $$7x^{3} + 5x^{2} + 8x + 10$$
The results of the operations are: 1. $$2x^{4} + 4x^{3} - 4x^{2} - 2x - 1$$ 2. $$-2x^{4} + x^{3} + x + 1$$ 3. $$7x^{3} + 5x^{2} + 8x + 10$$

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