Alright, let's solve each part of this question using the FOIL method. FOIL stands for First, Outer, Inner, Last, and it is a method used to multiply two binomials.
### a) [tex]\((x + 5)\)[/tex] and [tex]\((x + 3)\)[/tex]
Applying the FOIL method:
- First: [tex]\(x \cdot x = x^2\)[/tex]
- Outer: [tex]\(x \cdot 3 = 3x\)[/tex]
- Inner: [tex]\(5 \cdot x = 5x\)[/tex]
- Last: [tex]\(5 \cdot 3 = 15\)[/tex]
Combine all these terms:
[tex]\[ x^2 + 3x + 5x + 15 = x^2 + 8x + 15 \][/tex]
Result: [tex]\( x^2 + 8x + 15 \)[/tex]
### b) [tex]\((x + 5)\)[/tex] and [tex]\((x - 5)\)[/tex]
Applying the FOIL method:
- First: [tex]\(x \cdot x = x^2\)[/tex]
- Outer: [tex]\(x \cdot (-5) = -5x\)[/tex]
- Inner: [tex]\(5 \cdot x = 5x\)[/tex]
- Last: [tex]\(5 \cdot (-5) = -25\)[/tex]
Combine all these terms:
[tex]\[ x^2 - 5x + 5x - 25 = x^2 - 25 \][/tex]
Result: [tex]\( x^2 - 25 \)[/tex]
### c) [tex]\((2x - 9)\)[/tex] and [tex]\((4x - 7)\)[/tex]
Applying the FOIL method:
- First: [tex]\(2x \cdot 4x = 8x^2\)[/tex]
- Outer: [tex]\(2x \cdot (-7) = -14x\)[/tex]
- Inner: [tex]\(-9 \cdot 4x = -36x\)[/tex]
- Last: [tex]\(-9 \cdot (-7) = 63\)[/tex]
Combine all these terms:
[tex]\[ 8x^2 - 14x - 36x + 63 = 8x^2 - 50x + 63 \][/tex]
Result: [tex]\( 8x^2 - 50x + 63 \)[/tex]
### d) [tex]\((x - 5)\)[/tex] and [tex]\((x - 3)\)[/tex]
Applying the FOIL method:
- First: [tex]\(x \cdot x = x^2\)[/tex]
- Outer: [tex]\(x \cdot (-3) = -3x\)[/tex]
- Inner: [tex]\(-5 \cdot x = -5x\)[/tex]
- Last: [tex]\(-5 \cdot (-3) = 15\)[/tex]
Combine all these terms:
[tex]\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \][/tex]
Result: [tex]\( x^2 - 8x + 15 \)[/tex]
### e) [tex]\((7x - 5)\)[/tex] and [tex]\((4x + 3)\)[/tex]
Applying the FOIL method:
- First: [tex]\(7x \cdot 4x = 28x^2\)[/tex]
- Outer: [tex]\(7x \cdot 3 = 21x\)[/tex]
- Inner: [tex]\(-5 \cdot 4x = -20x\)[/tex]
- Last: [tex]\(-5 \cdot 3 = -15\)[/tex]
Combine all these terms:
[tex]\[ 28x^2 + 21x - 20x - 15 = 28x^2 + x - 15 \][/tex]
Result: [tex]\( 28x^2 + x - 15 \)[/tex]
So the final results for the given multiplications are:
a) [tex]\( x^2 + 8x + 15 \)[/tex]
b) [tex]\( x^2 - 25 \)[/tex]
c) [tex]\( 8x^2 - 50x + 63 \)[/tex]
d) [tex]\( x^2 - 8x + 15 \)[/tex]
e) [tex]\( 28x^2 + x - 15 \)[/tex]
