Certainly! Let's go through each part one-by-one.
(a) Multiply [tex]\((a + 3)\)[/tex] by [tex]\((a + 2)\)[/tex]:
[tex]\[
(a + 3)(a + 2) = a^2 + 2a + 3a + 6 = a^2 + 5a + 6
\][/tex]
(b) Multiply [tex]\((a + 2)\)[/tex] by [tex]\((a + 5)\)[/tex]:
[tex]\[
(a + 2)(a + 5) = a^2 + 5a + 2a + 10 = a^2 + 7a + 10
\][/tex]
(c) Considering [tex]\((x + 4)\)[/tex], we assume it is already in its simplest form:
[tex]\[
x + 4
\][/tex]
(d) Multiply [tex]\((x + 6)\)[/tex] by [tex]\((x + 9)\)[/tex]:
[tex]\[
(x + 6)(x + 9) = x^2 + 9x + 6x + 54 = x^2 + 15x + 54
\][/tex]
(e) Considering [tex]\((x - 7)\)[/tex], we assume it is already in its simplest form:
[tex]\[
x - 7
\][/tex]
(f) Considering [tex]\((x + 5)\)[/tex], we assume it is already in its simplest form:
[tex]\[
x + 5
\][/tex]
(g) Multiply [tex]\(3x + 8\)[/tex] by [tex]\(2x + 3\)[/tex]:
[tex]\[
(3x + 8)(2x + 3) = 3x \cdot 2x + 3x \cdot 3 + 8 \cdot 2x + 8 \cdot 3 = 6x^2 + 9x + 16x + 24 = 6x^2 + 25x + 24
\][/tex]
(h) Considering [tex]\((x + 2)\)[/tex], we assume it is already in its simplest form:
[tex]\[
x + 2
\][/tex]
(i) Multiply [tex]\((a + b)\)[/tex] by [tex]\((y + 1)\)[/tex]:
[tex]\[
(a + b)(y + 1) = a \cdot y + a \cdot 1 + b \cdot y + b \cdot 1 = ay + a + by + b = ay + a + by + b
\][/tex]
(j) Considering [tex]\((y + 1)\)[/tex], we assume it is already in its simplest form:
[tex]\[
y + 1
\][/tex]
(k) Multiply [tex]\(8x + 12\)[/tex] by [tex]\(x + 12\)[/tex]:
[tex]\[
(8x + 12)(x + 12) = 8x^2 + 96x + 12x + 144 = 8x^2 + 108x + 144
\][/tex]
(l) Multiply [tex]\((a - b)\)[/tex], [tex]\((xy - 3)\)[/tex], [tex]\((3x + 2)\)[/tex], and [tex]\((3x + 4)\)[/tex]:
[tex]\[
(a - b)(xy - 3)(3x + 2)(3x + 4)
\][/tex]
Expanding step-by-step can be quite lengthy and involves breaking down each part, but combining the results directly:
[tex]\[
9a x^3 y + 18a x^2 y - 27a x^2 + 8a x y - 54a x - 24a - 9b x^3 y - 18b x^2 y + 27b x^2 - 8b x y + 54b x + 24b
\][/tex]
To sum up, the results are:
1. (a) [tex]\(a^2 + 5a + 6\)[/tex]
2. (b) [tex]\(a^2 + 7a + 10\)[/tex]
3. (c) [tex]\(x + 4\)[/tex]
4. (d) [tex]\(x^2 + 15x + 54\)[/tex]
5. (e) [tex]\(x - 7\)[/tex]
6. (f) [tex]\(x + 5\)[/tex]
7. (g) [tex]\(6x^2 + 25x + 24\)[/tex]
8. (h) [tex]\(x + 2\)[/tex]
9. (i) [tex]\(ay + a + by + b\)[/tex] or rearranged [tex]\(a y + a + b y + b\)[/tex]
10. (j) [tex]\(y + 1\)[/tex]
11. (k) [tex]\(8x^2 + 108x + 144\)[/tex]
12. (l) [tex]\(9a x^3 y + 18a x^2 y - 27a x^2 + 8a x y - 54a x - 24a - 9b x^3 y - 18b x^2 y + 27b x^2 - 8b x y + 54b x + 24b\)[/tex]
