4a-3b+5c,-5a+4b-c
MathBot Answer:
The sum of \(4 a - 3 b + 5 c\) and \(-\left( 5 a \right) + 4 b - c\) is:
\[\begin{aligned}&=\left(4 a - 3 b + 5 c\right) + \left(-\left( 5 a \right) + 4 b - c\right) \\\\ &= \left(-1\right) 5 a + 4 a - 3 b + 4 b + 4 c \\\\ &= - a + b + 4 c \\\\\end{aligned}\]
The difference of \(4 a - 3 b + 5 c\) and \(-\left( 5 a \right) + 4 b - c\) is:
\[\begin{aligned}&=\left(4 a - 3 b + 5 c\right) - \left(-\left( 5 a \right) + 4 b - c\right) \\\\ &= 9 a - 4 b - 3 b + 6 c \\\\ &= 9 a - 7 b + 6 c \\\\\end{aligned}\]
The product of \(4 a - 3 b + 5 c\) and \(-\left( 5 a \right) + 4 b - c\) is:
\[\begin{aligned}&=\left(4 a - 3 b + 5 c\right) \cdot \left(-\left( 5 a \right) + 4 b - c\right) \\\\ &= \left(4 a - 3 b + 5 c\right) \left(- 5 a + 4 b - c\right) \\\\ &= - 20 a^{2} + 31 a b - 29 a c - 12 b^{2} + 23 b c - 5 c^{2} \\\\\end{aligned}\]
The quotient of \(4 a - 3 b + 5 c\) and \(-\left( 5 a \right) + 4 b - c\) is:
\[\begin{aligned}&= \frac{\left(4 a - 3 b + 5 c\right)}{\left(-\left( 5 a \right) + 4 b - c\right)} \\\\ &= \frac{4 a - 3 b + 5 c}{- 5 a + 4 b - c} \\\\ &= \frac{- 4 a + 3 b - 5 c}{5 a - 4 b + c} \\\\\end{aligned}\]