Divide `2x^(2)+x-5` by x+2 and verify the division algorithm.

Explanation

To divide the polynomial \(2x^2 + x - 5\) by \(x + 2\) and verify the division algorithm, we will follow these steps:

### Step 1: Set up the division
We want to divide \(2x^2 + x - 5\) (the dividend) by \(x + 2\) (the divisor).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(2x^2\) by the leading term of the divisor \(x\):
\[ \frac{2x^2}{x} = 2x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply the entire divisor \(x + 2\) by \(2x\):
\[ 2x(x + 2) = 2x^2 + 4x \]
Now, subtract this from the original polynomial:
\[ (2x^2 + x - 5) - (2x^2 + 4x) = x - 4x - 5 = -3x - 5 \]

### Step 4: Repeat the process
Now, we repeat the process with the new polynomial \(-3x - 5\).

1. Divide the leading term \(-3x\) by the leading term \(x\):
\[ \frac{-3x}{x} = -3 \]
This gives us the next term of the quotient.

2. Multiply the entire divisor \(x + 2\) by \(-3\):
\[ -3(x + 2) = -3x - 6 \]
3. Subtract this from \(-3x - 5\):
\[ (-3x - 5) - (-3x - 6) = -5 + 6 = 1 \]

### Step 5: Write the result
Now we have:
- Quotient: \(2x - 3\)
- Remainder: \(1\)

So we can express the division as:
\[ 2x^2 + x - 5 = (x + 2)(2x - 3) + 1 \]

### Step 6: Verify the division algorithm
According to the division algorithm:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
Substituting the values we found:
\[ 2x^2 + x - 5 = (x + 2)(2x - 3) + 1 \]

Now we will expand the right-hand side:
1. Multiply:
\[ (x + 2)(2x - 3) = 2x^2 - 3x + 4x - 6 = 2x^2 + x - 6 \]
2. Add the remainder:
\[ 2x^2 + x - 6 + 1 = 2x^2 + x - 5 \]

Since both sides are equal, we have verified the division algorithm.

### Final Result
- Quotient: \(2x - 3\)
- Remainder: \(1\)