Let's solve each equation one-by-one step-by-step.
### Problem 4: [tex]\(20abc(a + b - c)\)[/tex]
To solve this problem, we distribute each term inside the parenthesis by [tex]\(20abc\)[/tex]:
[tex]\[
20abc(a + b - c) = 20abc \cdot a + 20abc \cdot b - 20abc \cdot c
\][/tex]
So the solution is:
[tex]\[
20a^2bc + 20ab^2c - 20abc^2
\][/tex]
### Problem 5: [tex]\((x + y) \cdot (x^2 + y^2)\)[/tex]
To solve this problem, we distribute each term inside the parenthesis by [tex]\((x + y)\)[/tex]:
[tex]\[
(x + y) \cdot (x^2 + y^2) = x \cdot x^2 + x \cdot y^2 + y \cdot x^2 + y \cdot y^2
\][/tex]
So the solution is:
[tex]\[
x^3 + xy^2 + yx^2 + y^3
\][/tex]
### Problem 6: [tex]\((2a + b) \cdot (3a - 2b)\)[/tex]
To solve this problem, we distribute each term inside the parenthesis using the distributive property:
[tex]\[
(2a + b) \cdot (3a - 2b) = 2a \cdot 3a + 2a \cdot (-2b) + b \cdot 3a + b \cdot (-2b)
\][/tex]
Solving it step-by-step:
[tex]\[
= 6a^2 - 4ab + 3ab - 2b^2
\][/tex]
Combining like-terms, we get:
[tex]\[
= 6a^2 - ab - 2b^2
\][/tex]
### Problem 7: [tex]\((6a - 5b) \cdot (2b + 7a)\)[/tex]
To solve this problem, we distribute each term inside the parenthesis using the distributive property:
[tex]\[
(6a - 5b) \cdot (2b + 7a) = 6a \cdot 2b + 6a \cdot 7a - 5b \cdot 2b - 5b \cdot 7a
\][/tex]
Solving it step-by-step:
[tex]\[
= 12ab + 42a^2 - 10b^2 - 35ab
\][/tex]
Combining like-terms, we get:
[tex]\[
= 42a^2 - 23ab - 10b^2
\][/tex]
Substituting [tex]\(a = 4\)[/tex] and [tex]\(b = 6\)[/tex]:
[tex]\[
= 42(4)^2 - 23(4)(6) - 10(6)^2
\][/tex]
Calculating each term:
[tex]\[
= 42 \cdot 16 - 23 \cdot 24 - 10 \cdot 36
\][/tex]
[tex]\[
= 672 - 552 - 360
\][/tex]
[tex]\[
= -240
\][/tex]
### Problem 8: [tex]\((4x + y) \cdot (-2x - 5xy)\)[/tex]
To solve this problem, we distribute each term inside the parenthesis using the distributive property:
[tex]\[
(4x + y) \cdot (-2x - 5xy) = 4x \cdot (-2x) + 4x \cdot (-5xy) + y \cdot (-2x) + y \cdot (-5xy)
\][/tex]
Solving it step-by-step:
[tex]\[
= -8x^2 - 20x^2y - 2xy - 5xy^2
\][/tex]
Combining like-terms, we get:
[tex]\[
= -8x^2 - 20x^2y - 2xy - 5xy^2
\][/tex]
And that completes the detailed, step-by-step solutions for each problem.
