What must be added to p(x)`=4x^(4)-5x^(3)-39x^(2)-46x-2`, so that the resulting polynomial is divisible by g(x)`=4x^(2)+7x+2`?

Explanation

To solve the problem of what must be added to the polynomial \( p(x) = 4x^4 - 5x^3 - 39x^2 - 46x - 2 \) so that it is divisible by \( g(x) = 4x^2 + 7x + 2 \), we will follow these steps:

### Step 1: Understand the Division Algorithm
According to the polynomial division algorithm, we can express \( p(x) \) as:
\[ p(x) = g(x) \cdot q(x) + r(x) \]
where \( r(x) \) is the remainder when \( p(x) \) is divided by \( g(x) \). For \( p(x) \) to be divisible by \( g(x) \), the remainder \( r(x) \) must be zero.

### Step 2: Perform Polynomial Long Division
We will divide \( p(x) \) by \( g(x) \) to find the remainder \( r(x) \).

1. **Divide the leading term:**
Divide the leading term of \( p(x) \) by the leading term of \( g(x) \):
\[ \frac{4x^4}{4x^2} = x^2 \]
So, the first term of the quotient \( q(x) \) is \( x^2 \).

2. **Multiply and subtract:**
Multiply \( g(x) \) by \( x^2 \):
\[ x^2 \cdot (4x^2 + 7x + 2) = 4x^4 + 7x^3 + 2x^2 \]
Now subtract this from \( p(x) \):
\[ (4x^4 - 5x^3 - 39x^2 - 46x - 2) - (4x^4 + 7x^3 + 2x^2) = -12x^3 - 41x^2 - 46x - 2 \]

3. **Repeat the process:**
Now, divide the leading term of the new polynomial by the leading term of \( g(x) \):
\[ \frac{-12x^3}{4x^2} = -3x \]
Multiply \( g(x) \) by \( -3x \):
\[ -3x \cdot (4x^2 + 7x + 2) = -12x^3 - 21x^2 - 6x \]
Subtract:
\[ (-12x^3 - 41x^2 - 46x - 2) - (-12x^3 - 21x^2 - 6x) = -20x^2 - 40x - 2 \]

4. **Repeat again:**
Divide the leading term:
\[ \frac{-20x^2}{4x^2} = -5 \]
Multiply \( g(x) \) by \( -5 \):
\[ -5 \cdot (4x^2 + 7x + 2) = -20x^2 - 35x - 10 \]
Subtract:
\[ (-20x^2 - 40x - 2) - (-20x^2 - 35x - 10) = -5x + 8 \]

### Step 3: Identify the Remainder
The remainder \( r(x) \) is:
\[ r(x) = -5x + 8 \]

### Step 4: Determine What to Add
To make \( p(x) \) divisible by \( g(x) \), we need to add the negative of the remainder:
\[ \text{What to add} = -r(x) = -(-5x + 8) = 5x - 8 \]

### Final Answer
Thus, the polynomial that must be added to \( p(x) \) is:
\[ 5x - 8 \]