perform the operations and simplify completely -(5x^3+7x-8)+(3x-5)(-x^2+7x+2)

Answer 1

To perform the operations and simplify the expression \(- (5x^3 + 7x - 8) + (3x - 5)(-x^2 + 7x + 2)\), we will follow these steps:

1. Distribute the multiplication in \((3x - 5)(-x^2 + 7x + 2)\).
2. Simplify the expression.
3. Combine like terms.

Step 1: Multiply \((3x - 5)(-x^2 + 7x + 2)\)

Using distribution (the distributive property), we have: \[ (3x)(-x^2) + (3x)(7x) + (3x)(2) + (-5)(-x^2) + (-5)(7x) + (-5)(2) \]

Calculating each term:

1. \(3x \cdot -x^2 = -3x^3\)
2. \(3x \cdot 7x = 21x^2\)
3. \(3x \cdot 2 = 6x\)
4. \(-5 \cdot -x^2 = 5x^2\)
5. \(-5 \cdot 7x = -35x\)
6. \(-5 \cdot 2 = -10\)

Now, combining these: \[ -3x^3 + 21x^2 + 6x + 5x^2 - 35x - 10 \]

Combining like terms:

• The \(x^3\) term: \(-3x^3\)
• The \(x^2\) terms: \(21x^2 + 5x^2 = 26x^2\)
• The \(x\) terms: \(6x - 35x = -29x\)
• The constant term: \(-10\)

So, we get: \[ -3x^3 + 26x^2 - 29x - 10 \]

Step 2: Simplify \(- (5x^3 + 7x - 8)\)

Now apply the negative sign: \[ -(5x^3 + 7x - 8) = -5x^3 - 7x + 8 \]

Step 3: Combine both parts

Now we combine the results: \[

• (5x^3 + 7x - 8) + (3x - 5)(-x^2 + 7x + 2) = (-5x^3 - 7x + 8) + (-3x^3 + 26x^2 - 29x - 10) \]

Let's combine the like terms:

• The \(x^3\) terms: \(-5x^3 - 3x^3 = -8x^3\)
• The \(x^2\) term: \(26x^2\) (only one term)
• The \(x\) terms: \(-7x - 29x = -36x\)
• The constant terms: \(8 - 10 = -2\)

Final Result

Combining all of this, we arrive at the final simplified expression: \[ \boxed{-8x^3 + 26x^2 - 36x - 2} \]