Let's go through the questions step by step.

Part A: Find the remainder using the Remainder Theorem

1. $3x^5 + 5x^4 - 4x^3 + 7x + 3$, divided by $x + 2$:

   • According to the Remainder Theorem, the remainder of a polynomial $P(x)$ divided by $x - a$ can be found by evaluating $P(a)$.
   • Here, the divisor is $x + 2$, so we evaluate the polynomial at $x = -2$. $P(-2) = 3(-2)^5 + 5(-2)^4 - 4(-2)^3 + 7(-2) + 3$ Solving this gives the remainder.

2. $a^3 - 3a^2 - a + 20$, divided by $a + 2$:

   • Again, using the Remainder Theorem, evaluate the polynomial at $a = -2$. $P(-2) = (-2)^3 - 3(-2)^2 - (-2) + 20$ Solving this gives the remainder.

Part B: Check if the binomial is a factor using the Factor Theorem

1. $P(x) = x^3 + 4x^2 - 2x - 5$, checking $x + 1$:

   • The Factor Theorem states that if $x + 1$ is a factor, then $P(-1) = 0$. $P(-1) = (-1)^3 + 4(-1)^2 - 2(-1) - 5$ Simplifying this will show whether $P(-1) = 0$, confirming whether $x + 1$ is a factor.

2. $P(x) = 2x^3 + 9x^2 - 3x - 4$, checking $x + 4$:

   • Using the Factor Theorem, evaluate $P(-4)$: $P(-4) = 2(-4)^3 + 9(-4)^2 - 3(-4) - 4$ Simplify this expression to check if $P(-4) = 0$.

Would you like to see the full solutions for each of these problems, or do you have any specific questions?

Here are 5 related questions to expand your understanding:

1. How does the Remainder Theorem relate to synthetic division?
2. Can the Factor Theorem be applied to any polynomial?
3. What is the difference between the Remainder and Factor Theorems?
4. How would you solve the same problems using synthetic division instead?
5. What happens if you evaluate the polynomial at other values instead of the divisor's zero?

Tip: The Remainder Theorem is useful not only for finding remainders but also for checking potential factors through evaluation.