Question

1. divide.

\\(\frac{x^2 - 6x + 14}{x + 2}\\)
\\(\bigcirc\\) \\(x - 30 + \frac{5}{x + 2}\\)
\\(\bigcirc\\) \\(x + 4 + \frac{14}{x + 2}\\)
\\(\bigcirc\\) \\(x + 6 - \frac{32}{x + 2}\\)
\\(\bigcirc\\) \\(x - 8 + \frac{30}{x + 2}\\)

Explanation:

Step 1: Perform Polynomial Long Division

We divide \( x^2 - 6x + 14 \) by \( x + 2 \).
First, divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \), we get \( x \).
Multiply the divisor \( x + 2 \) by \( x \), we have \( x(x + 2)=x^2 + 2x \).
Subtract this from the dividend:
\( (x^2 - 6x + 14)-(x^2 + 2x)=x^2 - 6x + 14 - x^2 - 2x=-8x + 14 \)

Step 2: Continue the Division

Now, divide the leading term of \( -8x + 14 \) (which is \( -8x \)) by the leading term of the divisor \( x \), we get \( -8 \).
Multiply the divisor \( x + 2 \) by \( -8 \), we have \( -8(x + 2)=-8x - 16 \).
Subtract this from \( -8x + 14 \):
\( (-8x + 14)-(-8x - 16)=-8x + 14 + 8x + 16 = 30 \)

Step 3: Write the Result

So, when we divide \( x^2 - 6x + 14 \) by \( x + 2 \), we get the quotient \( x - 8 \) and the remainder \( 30 \).
In the form of \( \text{quotient}+\frac{\text{remainder}}{\text{divisor}} \), it is \( x - 8+\frac{30}{x + 2} \)

Answer:

\( x - 8 + \frac{30}{x + 2} \)