Sure, let's go through each part step-by-step:
### 2. Find the product.
#### (i) [tex]\((x^2 + 5x + 3)(x + 5)\)[/tex]
Distributing each term in [tex]\((x^2 + 5x + 3)\)[/tex] by each term in [tex]\((x + 5)\)[/tex]:
[tex]\[
(x + 5) (x^2 + 5x + 3)
\][/tex]
#### (ii) [tex]\((1 + x + x^2)(1 - 4x)\)[/tex]
Similarly, distribute each term in [tex]\((1 + x + x^2)\)[/tex] by each term in [tex]\((1 - 4x)\)[/tex]:
[tex]\[
(1 - 4x) (1 + x + x^2)
\][/tex]
#### (iii) [tex]\((x^2 + xy + y^2)(x^2 + y^2)\)[/tex]
Distribute each term in [tex]\((x^2 + xy + y^2)\)[/tex] by each term in [tex]\((x^2 + y^2)\)[/tex]:
[tex]\[
(x^2 + y^2) (x^2 + xy + y^2)
\][/tex]
#### (iv) [tex]\(\frac{1}{2}ab (3a^2 - 5b^2a + 10a^2b)\)[/tex]
Multiply each term in the polynomial by [tex]\(\frac{1}{2}ab\)[/tex]:
[tex]\[
a b \left(3 a^2 - 5 b^2 a + 10 a^2 b\right) / 2
\][/tex]
#### (v) [tex]\((2x^3 + 7x + 10x^2) \times \frac{5}{2} x\)[/tex]
Distribute [tex]\(\frac{5}{2} x\)[/tex] across each term in [tex]\((2x^3 + 7x + 10x^2)\)[/tex]:
[tex]\[
x (5 x^3 + 25 x^2 + 35 x / 2)
\][/tex]
### 3. Simplify
#### (i) [tex]\(\left(8x^3 - 4x^2 + 5\right) \times \frac{1}{2}x + \left(3x^2 - 5x + 1\right) \times 4x\)[/tex]
First, simplify each term individually:
[tex]\[
\frac{1}{2} x (8 x^3 - 4 x^2 + 5) = 4 x^4 - 2 x^3 + \frac{5}{2} x
\][/tex]
[tex]\[
4 x (3 x^2 - 5 x + 1) = 12 x^3 - 20 x^2 + 4 x
\][/tex]
Combine the terms:
[tex]\[
x (4 x^3 + 12 x^3 - 2 x^2 - 20 x + 4 x + \frac{5}{2})
\][/tex]
#### (ii) [tex]\((5a^3 - 18a^2 - 25a) \times \frac{1}{5}T (9a^2 - 6a + 11) \times \frac{-2}{3}a\)[/tex]
Simplify and combine terms:
[tex]\[
-\frac{2}{3} T a (1/a) (5 a^3 - 18 a^2 - 25 a) (9 a^2 - 6 a + 11)
\][/tex]
#### (iii) [tex]\((x^2 + xy + \frac{1}{2}xy^2) \times 2\pi + (y^2 + 2xy - x^2) \times \frac{3}{2}x\)[/tex]
Simplify each part separately:
[tex]\[
2 \pi (x^2 + xy + x y^2/2) = 2 \pi x^2 + 2 \pi x y + \pi x y^2
\][/tex]
[tex]\[
\frac{3}{2} x (y^2 + 2 x y - x^2) = \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3
\][/tex]
Combine the terms:
[tex]\[
\pi (2 x^2 + x y^2 + 2 x y) + x (- \frac{3}{2} x^2 + 3 x y + \frac{3}{2} y^2)
\][/tex]
#### (iv) [tex]\((9x^2 - 12xy + 4y^2) \times xy^2 - (4x^2 - 12y^2 + 9xy) \times x^2y\)[/tex]
Separate each term:
[tex]\[
(9 x^2 - 12 x y + 4 y^2) (x y^2) = 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4
\][/tex]
[tex]\[
(4 x^2 - 12 y^2 + 9 x y) (x^2 y) = 4 x^4 y - 12 x^2 y^3 + 9 x^3 y^2
\][/tex]
Combine the terms:
[tex]\[
x^2 y (9 x^2 y - 12 x^2 + 4 y^2 - (9 y^2 - 12 x^2 + 4 y^2))
\][/tex]
