Sure, let's tackle each part of the question step-by-step.
### Part (a)
Expand and simplify: [tex]\(5\left(4 x^2 - 2 x + 3\right)\)[/tex]
Step-by-step Solution:
1. Distribute the [tex]\(5\)[/tex] to each term inside the parentheses:
[tex]\[
5 \cdot 4 x^2 = 20 x^2
\][/tex]
[tex]\[
5 \cdot (-2 x) = -10 x
\][/tex]
[tex]\[
5 \cdot 3 = 15
\][/tex]
2. Combine the terms to get the expanded and simplified form:
[tex]\[
20 x^2 - 10 x + 15
\][/tex]
### Part (b)
Simplify: [tex]\(-4 x^2 + (-2 x^2)\)[/tex]
Step-by-step Solution:
1. Combine like terms:
[tex]\[
-4 x^2 - 2 x^2 = -6 x^2
\][/tex]
### Part (c)
Expand and simplify: [tex]\(2\left(x^2 + 4 x - 5\right) + \left(6 + x^2\right)\)[/tex]
Step-by-step Solution:
1. Distribute the [tex]\(2\)[/tex] in [tex]\(2(x^2 + 4x - 5)\)[/tex]:
[tex]\[
2 \cdot x^2 = 2 x^2
\][/tex]
[tex]\[
2 \cdot 4 x = 8 x
\][/tex]
[tex]\[
2 \cdot (-5) = -10
\][/tex]
2. Write everything together with the other terms:
[tex]\[
2 x^2 + 8 x - 10 + 6 + x^2
\][/tex]
3. Combine like terms:
[tex]\[
(2 x^2 + x^2) + 8 x + (-10 + 6)
\][/tex]
[tex]\[
3 x^2 + 8 x - 4
\][/tex]
### Part (d)
Simplify: [tex]\(3 x^2 + 4 x - x^2 - x\)[/tex]
Step-by-step Solution:
1. Combine like terms for [tex]\(x^2\)[/tex] terms:
[tex]\[
3 x^2 - x^2 = 2 x^2
\][/tex]
2. Combine like terms for [tex]\(x\)[/tex] terms:
[tex]\[
4 x - x = 3 x
\][/tex]
3. Write the simplified form:
[tex]\[
2 x^2 + 3 x
\][/tex]
### Part (e)
Simplify: [tex]\(7 x^3 - 3 x + 6 x^3 - x\)[/tex]
Step-by-step Solution:
1. Combine like terms for [tex]\(x^3\)[/tex] terms:
[tex]\[
7 x^3 + 6 x^3 = 13 x^3
\][/tex]
2. Combine like terms for [tex]\(x\)[/tex] terms:
[tex]\[
-3 x - x = -4 x
\][/tex]
3. Write the simplified form:
[tex]\[
13 x^3 - 4 x
\][/tex]
### Summary of Answers:
a) [tex]\(20 x^2 - 10 x + 15\)[/tex]
b) [tex]\(-6 x^2\)[/tex]
c) [tex]\(3 x^2 + 8 x - 4\)[/tex]
d) [tex]\(2 x^2 + 3 x\)[/tex]
e) [tex]\(13 x^3 - 4 x\)[/tex]
