To find the remainder when the polynomial $f(x) = 4x^2 - 3x + 5$ is divided by different binomials, we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial $f(x)$ is divided by $x - a$, the remainder is $f(a)$.

Let's calculate the remainder for each case:

1. When $f(x)$ is divided by $x + 2$: Using the Remainder Theorem, $f(-2)$: $f(-2) = 4(-2)^2 - 3(-2) + 5 = 4(4) + 6 + 5 = 16 + 6 + 5 = 27$ So, the remainder is $\mathbf{27}$.

2. When $f(x)$ is divided by $x + 1$: Using the Remainder Theorem, $f(-1)$: $f(-1) = 4(-1)^2 - 3(-1) + 5 = 4(1) + 3 + 5 = 4 + 3 + 5 = 12$ So, the remainder is $\mathbf{12}$.

3. When $f(x)$ is divided by $x$: Using the Remainder Theorem, $f(0)$: $f(0) = 4(0)^2 - 3(0) + 5 = 5$ So, the remainder is $\mathbf{5}$.

4. When $f(x)$ is divided by $x - 1$: Using the Remainder Theorem, $f(1)$: $f(1) = 4(1)^2 - 3(1) + 5 = 4(1) - 3 + 5 = 4 - 3 + 5 = 6$ So, the remainder is $\mathbf{6}$.

5. When $f(x)$ is divided by $x - 2$: Using the Remainder Theorem, $f(2)$: $f(2) = 4(2)^2 - 3(2) + 5 = 4(4) - 6 + 5 = 16 - 6 + 5 = 15$ So, the remainder is $\mathbf{15}$.

Summary of Remainders:

1. $x + 2$: remainder is 27
2. $x + 1$: remainder is 12
3. $x$: remainder is 5
4. $x - 1$: remainder is 6
5. $x - 2$: remainder is 15

Would you like more details on any of the steps or have any questions?

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Here are 5 related questions to explore further:

1. What is the Remainder Theorem, and how is it applied in polynomial division?
2. How would you factorize the polynomial $f(x) = 4x^2 - 3x + 5$?
3. How do you perform synthetic division to verify the remainders?
4. Can you determine the quotient when dividing $f(x)$ by $x + 2$?
5. How would you find the remainder if you divided $f(x)$ by a quadratic binomial?

Tip: The Remainder Theorem is a quick and efficient way to find remainders without needing to perform full polynomial division.