Let's solve each expression step-by-step.
1. Simplifying the Expression
[tex]$6xy^2 - 12x^2y = 6xyx^2 - 2x - 1xy$[/tex]
First, let’s simplify the entire expression.
Simplified version:
[tex]$x(-6x^2y - 12xy + 6y^2 + y + 2)$[/tex]
2. Solving the Equation
[tex]$-10n^3 - 15n^4 = -5n^3$[/tex]
Combining like terms and solving for [tex]\( n \)[/tex], we obtain the possible solutions for [tex]\( n \)[/tex]:
[tex]$n = -\frac{1}{3}, \quad n = 0$[/tex]
3. Simplifying another Expression
[tex]\[b^5c^3 + b^4c^6 - 3b^5c^5 - b^4c^3 \][/tex]
Simplified version:
[tex]$b^4c^3(-3bc^2 + b + c^3 - 1)$[/tex]
4. Simplifying another Expression
[tex]\[16xy^2 - 16xy - 4x - 4x \][/tex]
Simplified version:
[tex]$8x(2y^2 - 2y - 1)$[/tex]
5. Distributing and Combining Like Terms
[tex]\[(a + b)(x - 3) + (2a - b)(x - 3) - (x - 3)\][/tex]
Simplified version:
[tex]$3ax - 9a - x + 3$[/tex]
6. Simplifying another Expression by Combining Like Terms
[tex]\[x^2 + xy + xz + yz - (x + z)\][/tex]
Simplified version:
[tex]$x^2 + xy + xz - x + yz - z$[/tex]
7. Simplifying another Expression by Combining Like Terms
[tex]\[3p - mp + 3n - mn - (p + n)\][/tex]
Simplified version:
[tex]$-mn - mp + 2n + 2p$[/tex]
In summary, the detailed step-by-step solutions to the given expressions are:
1. [tex]\( x(-6x^2y - 12xy + 6y^2 + y + 2) \)[/tex]
2. Solutions for [tex]\( n \)[/tex] are [tex]\( n = -\frac{1}{3}, \)[/tex] [tex]\( n = 0 \)[/tex]
3. [tex]\( b^4c^3(-3bc^2 + b + c^3 - 1) \)[/tex]
4. [tex]\( 8x(2y^2 - 2y - 1) \)[/tex]
5. [tex]\( 3ax - 9a - x + 3 \)[/tex]
6. [tex]\( x^2 + xy + xz - x + yz - z \)[/tex]
7. [tex]\( -mn - mp + 2n + 2p \)[/tex]
These results match the output given.
