Question
Answer
$$(-x+5)(x^4+x^3+4x^2-2x-12)+(x-4)(x^3+6x^2+11x+30)=-x^5+5x^4+3x^3+9x^2-12x-180$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-x+5)(x^4+x^3+4x^2-2x-12)+(x-4)(x^3+6x^2+11x+30)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-x^5+4x^4+x^3+22x^2+2x-60+x^4+2x^3-13x^2-14x-120 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^5+5x^4+3x^3+9x^2-12x-180\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{-x+5}\right) $ by each term in $ \left( x^4+x^3+4x^2-2x-12\right) $. $$ \left( \color{blue}{-x+5}\right) \cdot \left( x^4+x^3+4x^2-2x-12\right) = -x^5-x^4-4x^3+2x^2+12x+5x^4+5x^3+20x^2-10x-60 $$ |
| ② | Combine like terms: $$ -x^5 \color{blue}{-x^4} \color{red}{-4x^3} + \color{green}{2x^2} + \color{orange}{12x} + \color{blue}{5x^4} + \color{red}{5x^3} + \color{green}{20x^2} \color{orange}{-10x} -60 = \\ = -x^5+ \color{blue}{4x^4} + \color{red}{x^3} + \color{green}{22x^2} + \color{orange}{2x} -60 $$Multiply each term of $ \left( \color{blue}{x-4}\right) $ by each term in $ \left( x^3+6x^2+11x+30\right) $. $$ \left( \color{blue}{x-4}\right) \cdot \left( x^3+6x^2+11x+30\right) = x^4+6x^3+11x^2+30x-4x^3-24x^2-44x-120 $$ |
| ③ | Combine like terms: $$ x^4+ \color{blue}{6x^3} + \color{red}{11x^2} + \color{green}{30x} \color{blue}{-4x^3} \color{red}{-24x^2} \color{green}{-44x} -120 = x^4+ \color{blue}{2x^3} \color{red}{-13x^2} \color{green}{-14x} -120 $$ |
| ④ | Combine like terms: $$ -x^5+ \color{blue}{4x^4} + \color{red}{x^3} + \color{green}{22x^2} + \color{orange}{2x} \color{blue}{-60} + \color{blue}{x^4} + \color{red}{2x^3} \color{green}{-13x^2} \color{orange}{-14x} \color{blue}{-120} = \\ = -x^5+ \color{blue}{5x^4} + \color{red}{3x^3} + \color{green}{9x^2} \color{orange}{-12x} \color{blue}{-180} $$ |
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