Sure! Let's complete the factored form of each given polynomial one by one:
1. Factor [tex]\(6xy^2 - 12x^2y\)[/tex]:
Notice that [tex]\(6xy^2\)[/tex] and [tex]\(-12x^2y\)[/tex] have a common factor of [tex]\(6xy\)[/tex]:
[tex]\[6xy^2 - 12x^2 y = 6xy(y - 2x)\][/tex]
2. Factor [tex]\(-10n^3 - 15n^4\)[/tex]:
Notice that [tex]\(-10n^3\)[/tex] and [tex]\(-15n^4\)[/tex] have a common factor of [tex]\(-5n^3\)[/tex]:
[tex]\[-10n^3 - 15n^4 = -5n^3(2 + 3n)\][/tex]
3. Factor [tex]\(b^5c^3 + b^4c^6 - 3b^5c^5\)[/tex]:
Notice that [tex]\(b^5c^3\)[/tex], [tex]\(b^4c^6\)[/tex], and [tex]\(-3b^5c^5\)[/tex] have a common factor of [tex]\(b^4c^3\)[/tex]:
[tex]\[b^5c^3 + b^4c^6 - 3b^5c^5 = b^4c^3(b + c^3 - 3bc^2)\][/tex]
4. Factor [tex]\(16xy^2 - 16xy - 4x\)[/tex]:
Notice that [tex]\(16xy^2\)[/tex], [tex]\(-16xy\)[/tex], and [tex]\(-4x\)[/tex] have a common factor of [tex]\(4x\)[/tex]:
[tex]\[16xy^2 - 16xy - 4x = 4x(4y^2 - 4y - 1)\][/tex]
5. Factor [tex]\((a+b)(x-3) + (2a-b)(x-3)\)[/tex]:
Notice that both terms have a common factor of [tex]\((x-3)\)[/tex]:
[tex]\[(a+b)(x-3) + (2a-b)(x-3) = (x-3)(a + b + 2a - b)\][/tex]
Simplify inside the parentheses:
[tex]\[(x-3)(3a)\][/tex]
6. Factor [tex]\(x^2 + xy + xz + yz\)[/tex]:
Group the terms:
[tex]\[x^2 + xy + xz + yz = (x^2 + xy) + (xz + yz)\][/tex]
Factor out the common factors:
[tex]\[= x(x + y) + z(x + y)\][/tex]
Notice that [tex]\((x + y)\)[/tex] is a common factor:
[tex]\[= (x + y)(x + z)\][/tex]
7. Factor [tex]\(3p - mp + 3n - mn\)[/tex]:
Group the terms:
[tex]\[3p - mp + 3n - mn = (3p - mp) + (3n - mn)\][/tex]
Factor out the common factors:
[tex]\[= p(3 - m) + n(3 - m)\][/tex]
Notice that [tex]\((3 - m)\)[/tex] is a common factor:
[tex]\[= (3 - m)(p + n)\][/tex]
So, the completed factored forms are:
1. [tex]\(6xy(y - 2x)\)[/tex]
2. [tex]\(-5n^3(2 + 3n)\)[/tex]
3. [tex]\(b^4c^3(b + c^3 - 3bc^2)\)[/tex]
4. [tex]\(4x(4y^2 - 4y - 1)\)[/tex]
5. [tex]\((x-3)(3a)\)[/tex]
6. [tex]\((x + y)(x + z)\)[/tex]
7. [tex]\((3 - m)(p + n)\)[/tex]
