Alright, let's work through the given expressions step-by-step to simplify them and find the respective answers based on known results:
### a) \((3x)(-2x^2 + 3)\)
1. Distribute \(3x\) to each term in the parenthesis:
[tex]\[
3x \cdot (-2x^2) + 3x \cdot 3
\][/tex]
2. Perform the multiplications:
[tex]\[
-6x^3 + 9x
\][/tex]
Thus, the simplified form of \((3x)(-2x^2 + 3)\) is:
[tex]\[
-6x^3 + 9x
\][/tex]
### b) \((6x^2)(6x^3 - 3x + 2)\)
1. Distribute \(6x^2\) to each term in the parenthesis:
[tex]\[
6x^2 \cdot 6x^3 + 6x^2 \cdot (-3x) + 6x^2 \cdot 2
\][/tex]
2. Perform the multiplications:
[tex]\[
36x^5 - 18x^3 + 12x^2
\][/tex]
Thus, the simplified form of \((6x^2)(6x^3 - 3x + 2)\) is:
[tex]\[
36x^5 - 18x^3 + 12x^2
\][/tex]
### c) \((12.2y)(10.5y - 3)\)
1. Distribute \(12.2y\) to each term in the parenthesis:
[tex]\[
12.2y \cdot 10.5y + 12.2y \cdot (-3)
\][/tex]
2. Perform the multiplications:
[tex]\[
128.1y^2 - 36.6y
\][/tex]
Thus, the simplified form of \((12.2y)(10.5y - 3)\) is:
[tex]\[
128.1y^2 - 36.6y
\][/tex]
### d) \((9.2m)(8.5m + 5)\)
1. Distribute \(9.2m\) to each term in the parenthesis:
[tex]\[
9.2m \cdot 8.5m + 9.2m \cdot 5
\][/tex]
2. Perform the multiplications:
[tex]\[
78.2m^2 + 46.0m
\][/tex]
Thus, the simplified form of \((9.2m)(8.5m + 5)\) is:
[tex]\[
78.2m^2 + 46.0m
\][/tex]
### e) \((22.3n)(20n^2 - 21n)\)
1. Distribute \(22.3n\) to each term in the parenthesis:
[tex]\[
22.3n \cdot 20n^2 + 22.3n \cdot (-21n)
\][/tex]
2. Perform the multiplications:
[tex]\[
446.0n^3 - 468.3n^2
\][/tex]
Thus, the simplified form of \((22.3n)(20n^2 - 21n)\) is:
[tex]\[
446.0n^3 - 468.3n^2
\][/tex]
These are the detailed step-by-step solutions for each given expression.
