If `m+ (1)/(m-2)=0`, find
`m^(5) + m^(4) + m^(3) + m^(2)+ m+1`= ?

Answer

To solve the equation \( m + \frac{1}{m - 2} = 0 \) and find the value of \( m^5 + m^4 + m^3 + m^2 + m + 1 \), we can follow these steps:

### Step 1: Solve for \( m \)
Start with the equation:
\[ m + \frac{1}{m - 2} = 0 \]
Rearranging gives:
\[ m = -\frac{1}{m - 2} \]

### Step 2: Eliminate the fraction
Multiply both sides by \( m - 2 \) (assuming \( m \neq 2 \)):
\[ m(m - 2) = -1 \]
Expanding this gives:
\[ m^2 - 2m + 1 = 0 \]

### Step 3: Factor the quadratic equation
The equation can be factored as:
\[ (m - 1)^2 = 0 \]
This means:
\[ m - 1 = 0 \implies m = 1 \]

### Step 4: Substitute \( m \) into the polynomial expression
Now we need to find:
\[ m^5 + m^4 + m^3 + m^2 + m + 1 \]
Substituting \( m = 1 \):
\[ 1^5 + 1^4 + 1^3 + 1^2 + 1 + 1 \]

### Step 5: Calculate the value
Calculating each term:
\[ 1 + 1 + 1 + 1 + 1 + 1 = 6 \]

### Final Answer
Thus, the value of \( m^5 + m^4 + m^3 + m^2 + m + 1 \) is:
\[ \boxed{6} \]