Solve for m : `((m + 2) ( 2 m - 3) - 2m^(2) + 6)/(m - 4)` = 7

Options

A. `(14)/(3)`
B. 10
C. 15
D. `(20)/(3)`

Answer

To solve the equation

\[ \frac{(m + 2)(2m - 3) - 2m^2 + 6}{m - 4} = 7, \]

we will follow these steps:

### Step 1: Multiply both sides by \(m - 4\)

To eliminate the denominator, we multiply both sides of the equation by \(m - 4\):

\[ (m + 2)(2m - 3) - 2m^2 + 6 = 7(m - 4). \]

### Step 2: Expand both sides

Now, we will expand both sides of the equation.

Left Side:
\[ (m + 2)(2m - 3) = m \cdot 2m + m \cdot (-3) + 2 \cdot 2m + 2 \cdot (-3) = 2m^2 - 3m + 4m - 6 = 2m^2 + m - 6. \]
So, the left side becomes:
\[ 2m^2 + m - 6 - 2m^2 + 6 = m. \]

Right Side:
\[ 7(m - 4) = 7m - 28. \]

Thus, the equation simplifies to:
\[ m = 7m - 28. \]

### Step 3: Rearrange the equation

Now, we will rearrange the equation to isolate \(m\):
\[ m - 7m = -28 \implies -6m = -28. \]

### Step 4: Solve for \(m\)

Now, divide both sides by -6:
\[ m = \frac{-28}{-6} = \frac{28}{6} = \frac{14}{3}. \]

### Final Answer

Thus, the solution for \(m\) is:
\[ m = \frac{14}{3}. \]

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