Alright, let's walk through each product step-by-step and find the expanded form.
### (a) [tex]\((x + 5)\)[/tex] and [tex]\((2x + 1)\)[/tex]
First, we use the distributive property (FOIL method):
[tex]\[ (x + 5)(2x + 1) = x \cdot 2x + x \cdot 1 + 5 \cdot 2x + 5 \cdot 1 \][/tex]
[tex]\[ = 2x^2 + x + 10x + 5 \][/tex]
Combine like terms:
[tex]\[ = 2x^2 + 11x + 5 \][/tex]
### (b) [tex]\((2x - 5)\)[/tex] and [tex]\((x + 1)\)[/tex]
Apply the distributive property:
[tex]\[ (2x - 5)(x + 1) = 2x \cdot x + 2x \cdot 1 - 5 \cdot x - 5 \cdot 1 \][/tex]
[tex]\[ = 2x^2 + 2x - 5x - 5 \][/tex]
Combine like terms:
[tex]\[ = 2x^2 - 3x - 5 \][/tex]
### (c) [tex]\((3x - 5)\)[/tex] and [tex]\((2x + 7)\)[/tex]
Apply the distributive property:
[tex]\[ (3x - 5)(2x + 7) = 3x \cdot 2x + 3x \cdot 7 - 5 \cdot 2x - 5 \cdot 7 \][/tex]
[tex]\[ = 6x^2 + 21x - 10x - 35 \][/tex]
Combine like terms:
[tex]\[ = 6x^2 + 11x - 35 \][/tex]
### (d) [tex]\((3x - 5y)\)[/tex] and [tex]\((2x + 5y)\)[/tex]
Apply the distributive property:
[tex]\[ (3x - 5y)(2x + 5y) = 3x \cdot 2x + 3x \cdot 5y - 5y \cdot 2x - 5y \cdot 5y \][/tex]
[tex]\[ = 6x^2 + 15xy - 10xy - 25y^2 \][/tex]
Combine like terms:
[tex]\[ = 6x^2 + 5xy - 25y^2 \][/tex]
### (e) [tex]\((3a + 6b)\)[/tex] and [tex]\((8a - 3b)\)[/tex]
Apply the distributive property:
[tex]\[ (3a + 6b)(8a - 3b) = 3a \cdot 8a + 3a \cdot (-3b) + 6b \cdot 8a + 6b \cdot (-3b) \][/tex]
[tex]\[ = 24a^2 - 9ab + 48ab - 18b^2 \][/tex]
Combine like terms:
[tex]\[ = 24a^2 + 39ab - 18b^2 \][/tex]
### (f) [tex]\((4x + 5y)\)[/tex] and [tex]\((2x - 3y)\)[/tex]
Apply the distributive property:
[tex]\[ (4x + 5y)(2x - 3y) = 4x \cdot 2x + 4x \cdot (-3y) + 5y \cdot 2x + 5y \cdot (-3y) \][/tex]
[tex]\[ = 8x^2 - 12xy + 10xy - 15y^2 \][/tex]
Combine like terms:
[tex]\[ = 8x^2 - 2xy - 15y^2 \][/tex]
### (g) [tex]\((5x + 4y)\)[/tex] and [tex]\((3x + 4y)\)[/tex]
Apply the distributive property:
[tex]\[ (5x + 4y)(3x + 4y) = 5x \cdot 3x + 5x \cdot 4y + 4y \cdot 3x + 4y \cdot 4y \][/tex]
[tex]\[ = 15x^2 + 20xy + 12xy + 16y^2 \][/tex]
Combine like terms:
[tex]\[ = 15x^2 + 32xy + 16y^2 \][/tex]
### (h) [tex]\((7x - 8y)\)[/tex] and [tex]\((5x - 6y)\)[/tex]
Apply the distributive property:
[tex]\[ (7x - 8y)(5x - 6y) = 7x \cdot 5x + 7x \cdot (-6y) - 8y \cdot 5x - 8y \cdot (-6y) \][/tex]
[tex]\[ = 35x^2 - 42xy - 40xy + 48y^2 \][/tex]
Combine like terms:
[tex]\[ = 35x^2 - 82xy + 48y^2 \][/tex]
Thus, the expanded forms for each product are:
(a) [tex]\(2x^2 + 11x + 5\)[/tex]
(b) [tex]\(2x^2 - 3x - 5\)[/tex]
(c) [tex]\(6x^2 + 11x - 35\)[/tex]
(d) [tex]\(6x^2 + 5xy - 25y^2\)[/tex]
(e) [tex]\(24a^2 + 39ab - 18b^2\)[/tex]
(f) [tex]\(8x^2 - 2xy - 15y^2\)[/tex]
(g) [tex]\(15x^2 + 32xy + 16y^2\)[/tex]
(h) [tex]\(35x^2 - 82xy + 48y^2\)[/tex]
