Given `3x^(2) +x=1`, find the value of `6x^(3) - x^(2) -3x + 2010`.

Answer

To solve the equation \(3x^2 + x = 1\) and find the value of \(6x^3 - x^2 - 3x + 2010\), we can follow these steps:

### Step 1: Rearrange the given equation
We start with the equation:
\[ 3x^2 + x - 1 = 0 \]

### Step 2: Express \(6x^3\) in terms of \(3x^2 + x\)
We know that:
\[ 6x^3 = 2x(3x^2) \]
From the rearranged equation \(3x^2 = 1 - x\), we can substitute \(3x^2\) in the expression for \(6x^3\):
\[ 6x^3 = 2x(1 - x) = 2x - 2x^2 \]

### Step 3: Substitute \(6x^3\) into the expression
Now we substitute \(6x^3\) into the expression \(6x^3 - x^2 - 3x + 2010\):
\[ 6x^3 - x^2 - 3x + 2010 = (2x - 2x^2) - x^2 - 3x + 2010 \]

### Step 4: Combine like terms
Now we combine the terms:
\[ = 2x - 2x^2 - x^2 - 3x + 2010 \]
\[ = -3x^2 - x + 2010 \]

### Step 5: Substitute \(3x^2\) from the original equation
Using the original equation \(3x^2 + x - 1 = 0\), we can express \(3x^2\) as:
\[ 3x^2 = 1 - x \]
Now substitute this into our expression:
\[ -3x^2 - x + 2010 = -(1 - x) - x + 2010 \]
\[ = -1 + x - x + 2010 \]
\[ = 2009 \]

### Final Result
Thus, the value of \(6x^3 - x^2 - 3x + 2010\) is:
\[ \boxed{2009} \]