Find the dimensions of cube whose volume is given by the expression. `4x^(2)+14x+6`.

Answer

To find the dimensions of a cube whose volume is given by the expression \(4x^2 + 14x + 6\), we can follow these steps:

### Step 1: Write down the volume expression
The volume \(V\) of the cube is given as:
\[ V = 4x^2 + 14x + 6 \]

### Step 2: Factor out the common factor
We notice that all terms in the expression can be divided by 2. So, we factor out 2:
\[ V = 2(2x^2 + 7x + 3) \]

### Step 3: Factor the quadratic expression
Next, we need to factor the quadratic expression \(2x^2 + 7x + 3\). We will split the middle term. We need two numbers that multiply to \(2 \times 3 = 6\) and add up to \(7\). These numbers are \(6\) and \(1\).

So, we can rewrite the expression:
\[ 2x^2 + 6x + 1x + 3 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (2x^2 + 6x) + (1x + 3) \]

### Step 5: Factor by grouping
From the first group \(2x^2 + 6x\), we can factor out \(2x\):
\[ 2x(x + 3) \]
From the second group \(1x + 3\), we can factor out \(1\):
\[ 1(x + 3) \]
Now we can combine these:
\[ 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) \]

### Step 6: Write the complete factorization of the volume
So, the volume can be expressed as:
\[ V = 2(2x + 1)(x + 3) \]

### Step 7: Identify the dimensions
Since the volume of a cube is given by \(V = a^3\) where \(a\) is the length of a side, we can equate:
\[ a = \sqrt[3]{2(2x + 1)(x + 3)} \]
However, since we are looking for the dimensions, we can take the factors directly as dimensions:
- One dimension is \(2\)
- The second dimension is \(2x + 1\)
- The third dimension is \(x + 3\)

Thus, the dimensions of the cube are:
\[ \text{Dimensions} = 2, \, (2x + 1), \, (x + 3) \]