long division (x2+3x-10) $$ \frac{\begin{pmatrix}x-2\\ \\ \end{pmatrix}}{\placeholder{}} $$

To perform the long division of $ \frac{x^2 + 3x - 10}{x - 2} $, we follow these steps:

### Step 1: Divide the first term of the dividend by the first term of the divisor
The first term of the dividend is $ x^2 $ and the first term of the divisor is $ x $. Divide:

\[
\frac{x^2}{x} = x
\]

### Step 2: Multiply the divisor by the result
Now, multiply the entire divisor $ x - 2 $ by $ x $:

\[
x(x - 2) = x^2 - 2x
\]

### Step 3: Subtract this result from the dividend
Now subtract $ (x^2 - 2x) $ from the original polynomial $ (x^2 + 3x - 10) $:

\[
(x^2 + 3x - 10) - (x^2 - 2x) = 5x - 10
\]

### Step 4: Repeat the process
Now, divide the first term of the new expression $ 5x $ by the first term of the divisor $ x $:

\[
\frac{5x}{x} = 5
\]

### Step 5: Multiply the divisor by the result
Multiply $ x - 2 $ by $ 5 $:

\[
5(x - 2) = 5x - 10
\]

### Step 6: Subtract this from the current expression
Now subtract $ (5x - 10) $ from $ (5x - 10) $:

\[
(5x - 10) - (5x - 10) = 0
\]

Since we have no remainder, the quotient is:

\[
x + 5
\]

### Final Answer:
\[
\frac{x^2 + 3x - 10}{x - 2} = x + 5
\]

Let me know if you'd like any clarification or further details on this.

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Here are some related questions you might find helpful:
1. How does long division work when dividing polynomials?
2. What happens when there is a remainder in polynomial division?
3. Can synthetic division be used to divide $ x^2 + 3x - 10 $ by $ x - 2 $?
4. How do you handle division by a quadratic or higher-degree polynomial?
5. Why is the degree of the divisor important in long division?

**Tip**: When performing long division with polynomials, always align terms by their degree to avoid confusion while subtracting.

long division (x^2 + 3x - 10) divided by (x - 2)

Learn how to perform long division with polynomials through this example of dividing (x^2 + 3x - 10) by (x - 2). This step-by-step solution covers key algebraic concepts and techniques. Suitable for students in Grades 9-12.

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