Solve `x^4 +x^3- 16x^2 -4x + 48 =0` given that the product of two of the roots is 6.

Answer

To solve the equation \(x^4 + x^3 - 16x^2 - 4x + 48 = 0\) given that the product of two of the roots is 6, we will follow these steps:

### Step 1: Identify Possible Roots
Since the product of two roots is 6, the possible pairs of roots could be (2, 3) or (-2, -3). We will start by checking if 2 is a root of the polynomial.

### Step 2: Synthetic Division by 2
We perform synthetic division of the polynomial by \(x - 2\):

\[ \begin{array}{r|rrrrr} 2 & 1 & 1 & -16 & -4 & 48 \\ & & 2 & 6 & -20 & -8 \\ \hline & 1 & 3 & -10 & -24 & 0 \\ \end{array} \]

The result of the division is \(x^3 + 3x^2 - 10x - 24\).

### Step 3: Check for Another Root (-2)
Next, we will check if -2 is a root of the resulting polynomial \(x^3 + 3x^2 - 10x - 24\) using synthetic division:

\[ \begin{array}{r|rrrr} -2 & 1 & 3 & -10 & -24 \\ & & -2 & -2 & 24 \\ \hline & 1 & 1 & -12 & 0 \\ \end{array} \]

The result of this division is \(x^2 + x - 12\).

### Step 4: Factor the Quadratic
Now we will factor \(x^2 + x - 12\):

\[ x^2 + x - 12 = (x + 4)(x - 3) \]

### Step 5: Find All Roots
Now we can find all the roots of the original polynomial:

1. From \(x - 2 = 0\), we get \(x = 2\).
2. From \(x + 2 = 0\), we get \(x = -2\).
3. From \(x + 4 = 0\), we get \(x = -4\).
4. From \(x - 3 = 0\), we get \(x = 3\).

### Final Roots
Thus, the roots of the equation \(x^4 + x^3 - 16x^2 - 4x + 48 = 0\) are:

\[ \boxed{2, -2, 3, -4} \]

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