Divide p(x) by g(x) in each of the following questions and find the quotient q(x) and remainder r(x) :
`p(x)=x^(3)+6x^(2)-1, " " g(x)=x^(2)+2`

Answer

To divide the polynomial \( p(x) = x^3 + 6x^2 - 1 \) by \( g(x) = x^2 + 2 \), we will perform polynomial long division. Here are the steps:

### Step 1: Set up the division
Write \( p(x) \) (the dividend) and \( g(x) \) (the divisor) in a long division format:

\[ \begin{array}{r|l} x^2 + 2 & x^3 + 6x^2 + 0x - 1 \\ \end{array} \]

### Step 2: Divide the leading terms
Divide the leading term of \( p(x) \) by the leading term of \( g(x) \):

\[ \frac{x^3}{x^2} = x \]

This gives us the first term of the quotient \( q(x) \).

### Step 3: Multiply and subtract
Multiply \( g(x) \) by \( x \) and subtract from \( p(x) \):

\[ x \cdot (x^2 + 2) = x^3 + 2x \]

Now subtract this from \( p(x) \):

\[ (x^3 + 6x^2 - 1) - (x^3 + 2x) = 6x^2 - 2x - 1 \]

### Step 4: Repeat the process
Now, we repeat the process with the new polynomial \( 6x^2 - 2x - 1 \).

Divide the leading term of \( 6x^2 \) by the leading term of \( g(x) \):

\[ \frac{6x^2}{x^2} = 6 \]

Add this to the quotient \( q(x) \):

\[ q(x) = x + 6 \]

### Step 5: Multiply and subtract again
Multiply \( g(x) \) by \( 6 \):

\[ 6 \cdot (x^2 + 2) = 6x^2 + 12 \]

Now subtract this from \( 6x^2 - 2x - 1 \):

\[ (6x^2 - 2x - 1) - (6x^2 + 12) = -2x - 1 - 12 = -2x - 13 \]

### Step 6: Determine the remainder
Now we have:

\[ \text{Remainder } r(x) = -2x - 13 \]

### Final Result
Thus, the quotient \( q(x) \) and remainder \( r(x) \) are:

\[ q(x) = x + 6 \]
\[ r(x) = -2x - 13 \]