Given that f(x) = x3 + 2x2 + 3x, find the remainder when f(x + 2) isdivided by x − 2.

To find the remainder when f(x + 2) is divided by x - 2, we first need to substitute (x + 2) into the function f(x).

So, f(x + 2) = (x + 2)³ + 2(x + 2)² + 3(x + 2)

Simplify this to get: f(x + 2) = x³ + 6x² + 12x + 8 + 2x² + 8x + 8 + 3x + 6

Combine like terms to get: f(x + 2) = x³ + 8x² + 23x + 22

Now, we want to find the remainder when f(x + 2) is divided by x - 2.

We can use the Remainder Theorem, which states that the remainder when a polynomial f(x) is divided by (x - a) is equal to f(a).

So, we substitute x = 2 into f(x + 2) to find the remainder:

Remainder = f(2) = 2³ + 82² + 232 + 22 = 8 + 32 + 46 + 22 = 108

So, the remainder when f(x + 2) is divided by x - 2 is 108.

First, we need to substitute (x+2) into the function f(x):

f(x+2) = (x+2)^3 + 2(x+2)^2 + 3(x+2)

Simplify this to get:

f(x+2) = x^3 + 6x^2 + 12x + 8 + 2x^2 + 8x + 8 + 3x + 6

Combine like terms:

f(x+2) = x^3 + 8x^2 + 23x + 22

Now, we want to find the remainder when f(x+2) is divided by (x-2).

We can use the Remainder Theorem, which states that the remainder when a polynomial f(x) is divided by (x-a) is equal to f(a).

So, we substitute x = 2 into f(x+2):

f(2+2) = (2+2)^3 + 8(2+2)^2 + 23(2+2) + 22

Simplify this to get:

f(4) = 4^3 + 84^2 + 234 + 22 = 64 + 128 + 92 + 22 = 306

So, the remainder when f(x+2) is divided by (x-2) is 306.