Sure, let's analyze each of the polynomial products to determine which ones result in a difference of squares or a perfect square trinomial.
1. [tex]\((5x + 3)(5x - 3)\)[/tex]
- This fits the form [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex].
- [tex]\(a = 5x\)[/tex] and [tex]\(b = 3\)[/tex].
- Therefore, this is a difference of squares: result is [tex]\(25x^2 - 9\)[/tex].
2. [tex]\((7x + 4)(7x + 4)\)[/tex]
- This fits the form [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
- [tex]\(a = 7x\)[/tex] and [tex]\(b = 4\)[/tex].
- Therefore, this is a perfect square trinomial: result is [tex]\(49x^2 + 56x + 16\)[/tex].
3. [tex]\((2x + 1)(x + 2)\)[/tex]
- This does not fit either form (neither difference of squares nor perfect square trinomial).
- Expand this to get: [tex]\(2x^2 + 4x + x + 2 = 2x^2 + 5x + 2\)[/tex].
4. [tex]\((4x - 6)(x + 8)\)[/tex]
- This does not fit either form (neither difference of squares nor perfect square trinomial).
- Expand this to get: [tex]\(4x^2 + 32x - 6x - 48 = 4x^2 + 26x - 48\)[/tex].
5. [tex]\((x - 9)(x - 9)\)[/tex]
- This fits the form [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
- [tex]\(a = x\)[/tex] and [tex]\(b = 9\)[/tex].
- Therefore, this is a perfect square trinomial: result is [tex]\(x^2 - 18x + 81\)[/tex].
6. [tex]\((-3x - 6)(-3x + 6)\)[/tex]
- This fits the form [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex].
- [tex]\(a = -3x\)[/tex] and [tex]\(b = 6\)[/tex].
- Therefore, this is a difference of squares: result is [tex]\(9x^2 - 36\)[/tex].
So, the products that result in a difference of squares or a perfect square trinomial are:
- [tex]\((5x + 3)(5x - 3)\)[/tex]
- [tex]\((7x + 4)(7x + 4)\)[/tex]
- [tex]\((x - 9)(x - 9)\)[/tex]
- [tex]\((-3x - 6)(-3x + 6)\)[/tex]
