Divide 3m minus 4 plus 2m squared by m plus 5.

Step-by-step solution

1. 1

   Rewrite in descending powers. $3m-4+2m^2 = 2m^2+3m-4$, giving the coefficient list $2,\ 3,\ -4$.

2. 2

   Pick the root from the divisor. $m+5=0$ at $m=-5$, so the synthetic-division multiplier is $-5$. Using $+5$ here is the most frequent sign error.

3. 3

   Perform the division. Bring down $2$; $2\times(-5)=-10$ and $3+(-10)=-7$; $-7\times(-5)=35$ and $-4+35=31$. The row is $2,\ -7,\ 31$.

4. 4

   Write the result. $\frac{2m^2+3m-4}{m+5} = 2m-7+\frac{31}{m+5}.$ Note how the negative root makes the middle coefficient flip sign relative to the numerator.

5. 5

   Check with the remainder theorem. At $m=-5$ the numerator is $2(25)+3(-5)-4 = 50-15-4 = 31$, exactly the remainder.

6. 6

   Verify by expansion. $(m+5)(2m-7)+31 = 2m^2-7m+10m-35+31 = 2m^2+3m-4$, which is the original numerator.