Sure, let's go through the process of performing the operations for each given expression.
### Expression 1: [tex]\((x+3)(x^2 - 2x + 5)\)[/tex]
To expand [tex]\((x+3)(x^2 - 2x + 5)\)[/tex], we'll use the distributive property:
[tex]\[
(x + 3)(x^2 - 2x + 5) = x(x^2 - 2x + 5) + 3(x^2 - 2x + 5)
\][/tex]
[tex]\[
= x^3 - 2x^2 + 5x + 3x^2 - 6x + 15
\][/tex]
Combine like terms:
[tex]\[
= x^3 + (3x^2 - 2x^2) + (5x - 6x) + 15
\][/tex]
[tex]\[
= x^3 + x^2 - x + 15
\][/tex]
So, the expanded form of [tex]\((x + 3)(x^2 - 2x + 5)\)[/tex] is:
[tex]\[
x^3 + x^2 - x + 15
\][/tex]
### Expression 2: [tex]\(x^3 + x^2 - x + 15\)[/tex]
This expression is already simplified and doesn't require any further operations. It remains:
[tex]\[
x^3 + x^2 - x + 15
\][/tex]
### Expression 3: [tex]\(x^4 + 15\)[/tex]
This expression is also already in its simplest form:
[tex]\[
x^4 + 15
\][/tex]
### Expression 4: [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3\)[/tex]
First, expand [tex]\((x + 3)^2\)[/tex]:
[tex]\[
(x + 3)^2 = (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9
\][/tex]
Now, distribute [tex]\(-2\)[/tex] to [tex]\((x+3)\)[/tex]:
[tex]\[
-2(x + 3) = -2x - 6
\][/tex]
Now, put all the pieces together:
[tex]\[
(x + 3)^2 - 2(x + 3) + 5x + 3 \to x^2 + 6x + 9 - 2x - 6 + 5x + 3
\][/tex]
Combine like terms:
[tex]\[
x^2 + (6x - 2x + 5x) + (9 - 6 + 3)
\][/tex]
[tex]\[
= x^2 + 9x + 6
\][/tex]
So, the simplified form of [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3\)[/tex] is:
[tex]\[
x^2 + 9x + 6
\][/tex]
To summarize, the results of the operations are:
1. [tex]\((x + 3)(x^2 - 2x + 5) = x^3 + x^2 - x + 15\)[/tex]
2. [tex]\(x^3 + x^2 - x + 15\)[/tex] (already simplified)
3. [tex]\(x^4 + 15\)[/tex] (already simplified)
4. [tex]\((x + 3)^2 - 2(x + 3) + 5x + 3 = x^2 + 9x + 6\)[/tex]
