Divide negative 5p plus 4p squared plus 4 by p minus 2.

Step-by-step solution

1. 1

   Reorder the numerator. $-5p+4p^2+4 = 4p^2-5p+4$, so the coefficients in descending order are $4,\ -5,\ 4$. Scrambled term order is deliberate in these exercises - sorting first is non-negotiable.

2. 2

   Identify the root. $p-2=0$ gives $p=2$.

3. 3

   Divide synthetically. Bring down $4$; $4\times2=8$ and $-5+8=3$; $3\times2=6$ and $4+6=10$. The bottom row is $4,\ 3,\ 10$.

4. 4

   Read the quotient and remainder. $\frac{4p^2-5p+4}{p-2} = 4p+3+\frac{10}{p-2}.$ A nonzero remainder means $p-2$ is not a factor of the numerator.

5. 5

   Cross-check with the remainder theorem. The numerator at $p=2$ is $4(4)-5(2)+4 = 16-10+4 = 10$, matching the remainder.

6. 6

   Verify the identity. $(p-2)(4p+3)+10 = 4p^2+3p-8p-6+10 = 4p^2-5p+4$, which is the original numerator.