Divide 4p minus 2 plus 3p squared by p minus 1.

Step-by-step solution

1. 1

   Put the numerator in descending order first. $4p-2+3p^2$ becomes $3p^2+4p-2$. Synthetic division reads coefficients strictly from the highest power down, so an out-of-order polynomial must be rewritten before anything else.

2. 2

   Set up with the root p = 1. The divisor $p-1$ vanishes at $p=1$, and the coefficient list is $3,\ 4,\ -2$.

3. 3

   Run the division. Bring down $3$; $3\times1=3$ and $4+3=7$; $7\times1=7$ and $-2+7=5$. The bottom row is $3,\ 7,\ 5$.

4. 4

   Interpret the row. The first two entries are the quotient $3p+7$ (one degree lower than the numerator), and the last entry $5$ is the remainder: $\frac{3p^2+4p-2}{p-1} = 3p+7+\frac{5}{p-1}.$

5. 5

   Confirm the remainder with the remainder theorem. Evaluating the numerator at $p=1$: $3(1)+4(1)-2 = 5$, which matches the remainder exactly - an independent check that costs one substitution.

6. 6

   Verify the whole identity. $(p-1)(3p+7)+5 = 3p^2+7p-3p-7+5 = 3p^2+4p-2$, the original numerator.