Add: `5x^(2) - 7x + 3, -8x^(2) + 2x -5` and `7x^(2) - x - 2`

Options

A. `4x^(2) + 6x -4`
B. `4x^(2) - 6x -4`
C. `4x^(2) - 6x +4`
D. `4x^(2) + 6x +4`

Answer

To solve the problem of adding the three algebraic expressions \(5x^{2} - 7x + 3\), \(-8x^{2} + 2x - 5\), and \(7x^{2} - x - 2\), we will follow these steps:

### Step 1: Write down the expressions
We start by writing down the three expressions clearly:
1. \(5x^{2} - 7x + 3\)
2. \(-8x^{2} + 2x - 5\)
3. \(7x^{2} - x - 2\)

### Step 2: Combine like terms
Next, we will combine the like terms from all three expressions. Like terms are terms that have the same variable raised to the same power.

- **Quadratic terms** (\(x^{2}\)):
- From the first expression: \(5x^{2}\)
- From the second expression: \(-8x^{2}\)
- From the third expression: \(7x^{2}\)

Combining these:
\[ 5x^{2} - 8x^{2} + 7x^{2} = (5 - 8 + 7)x^{2} = 4x^{2} \]

- **Linear terms** (\(x\)):
- From the first expression: \(-7x\)
- From the second expression: \(2x\)
- From the third expression: \(-x\)

Combining these:
\[ -7x + 2x - x = (-7 + 2 - 1)x = -6x \]

- **Constant terms**:
- From the first expression: \(3\)
- From the second expression: \(-5\)
- From the third expression: \(-2\)

Combining these:
\[ 3 - 5 - 2 = 3 - 7 = -4 \]

### Step 3: Write the final expression
Now, we can write the final expression by combining all the results from the like terms:
\[ 4x^{2} - 6x - 4 \]

### Final Answer
The sum of the three algebraic expressions is:
\[ \boxed{4x^{2} - 6x - 4} \]

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