Subtract `6x^(3) -5x^(2)+4x-3` from the sum `x+2x^(2)-3x^(3)` and `2-x^(2)+6x-x^(3)`.

Answer

To solve the problem step by step, we need to follow the instructions given in the question. Let's break it down:

### Step 1: Find the sum of the two expressions
We need to first add the two algebraic expressions:
1. \( x + 2x^2 - 3x^3 \)
2. \( 2 - x^2 + 6x - x^3 \)

**Combine like terms:**
- The \(x^3\) terms: \(-3x^3 - x^3 = -4x^3\)
- The \(x^2\) terms: \(2x^2 - x^2 = 1x^2\)
- The \(x\) terms: \(x + 6x = 7x\)
- The constant terms: \(2\)

So, the sum is:
\[ -4x^3 + x^2 + 7x + 2 \]

### Step 2: Subtract the first expression from the sum
Now we need to subtract the expression \(6x^3 - 5x^2 + 4x - 3\) from the sum we just calculated.

**Write the subtraction:**
\[ (-4x^3 + x^2 + 7x + 2) - (6x^3 - 5x^2 + 4x - 3) \]

### Step 3: Change the signs of the second expression
When we subtract, we change the signs of the second expression:
\[ -4x^3 + x^2 + 7x + 2 - 6x^3 + 5x^2 - 4x + 3 \]

### Step 4: Combine like terms again
Now, we combine the like terms:
- The \(x^3\) terms: \(-4x^3 - 6x^3 = -10x^3\)
- The \(x^2\) terms: \(x^2 + 5x^2 = 6x^2\)
- The \(x\) terms: \(7x - 4x = 3x\)
- The constant terms: \(2 + 3 = 5\)

So, the final result after combining all the terms is:
\[ -10x^3 + 6x^2 + 3x + 5 \]

### Final Answer:
The final answer is:
\[ -10x^3 + 6x^2 + 3x + 5 \]

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