Suppose that the function f(x) and g(x) satisfy the system of equations `f(x)+3g(x)=x^(2)+x+6`
and `2f(x)+4g(x)=2x^(2)+4` for every x. The value of `x` for which `f(x)=g(x)` can be equal to

Options

A. `-2`
B. `2`
C. `-5`
D. `5`

Explanation

To solve the problem, we need to find the values of \( x \) for which \( f(x) = g(x) \) given the equations:

1. \( f(x) + 3g(x) = x^2 + x + 6 \) (Equation 1)
2. \( 2f(x) + 4g(x) = 2x^2 + 4 \) (Equation 2)

### Step 1: Simplify the equations

First, we can rewrite Equation 2 by dividing everything by 2:

\[ f(x) + 2g(x) = x^2 + 2 \quad \text{(Equation 3)} \]

### Step 2: Set up a system of equations

Now we have two equations to work with:

1. \( f(x) + 3g(x) = x^2 + x + 6 \) (Equation 1)
2. \( f(x) + 2g(x) = x^2 + 2 \) (Equation 3)

### Step 3: Eliminate \( f(x) \)

Now, we can eliminate \( f(x) \) by subtracting Equation 3 from Equation 1:

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

This simplifies to:

\[ g(x) = x + 4 \quad \text{(Equation 4)} \]

### Step 4: Substitute \( g(x) \) back into one of the original equations

Now that we have \( g(x) \), we can substitute it back into Equation 3 to find \( f(x) \):

\[ f(x) + 2(x + 4) = x^2 + 2 \]

Expanding this gives:

\[ f(x) + 2x + 8 = x^2 + 2 \]

Now, isolate \( f(x) \):

\[ f(x) = x^2 + 2 - 2x - 8 \]

This simplifies to:

\[ f(x) = x^2 - 2x - 6 \quad \text{(Equation 5)} \]

### Step 5: Set \( f(x) = g(x) \)

Now we need to find the values of \( x \) for which \( f(x) = g(x) \):

\[ x^2 - 2x - 6 = x + 4 \]

### Step 6: Rearrange the equation

Rearranging gives:

\[ x^2 - 2x - x - 6 - 4 = 0 \]

This simplifies to:

\[ x^2 - 3x - 10 = 0 \]

### Step 7: Factor the quadratic equation

Now we can factor the quadratic:

\[ (x - 5)(x + 2) = 0 \]

### Step 8: Solve for \( x \)

Setting each factor to zero gives us:

\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Conclusion

The values of \( x \) for which \( f(x) = g(x) \) are:

\[ \boxed{-2 \text{ and } 5} \]