Tap the blue circles to see an explanation.
$$ \begin{aligned}(x+9)(x+4)(x+1)(x-5)(x-11)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+4x+9x+36)(x+1)(x-5)(x-11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2+13x+36)(x+1)(x-5)(x-11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+x^2+13x^2+13x+36x+36)(x-5)(x-11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3+14x^2+49x+36)(x-5)(x-11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4+9x^3-21x^2-209x-180)(x-11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^5-2x^4-120x^3+22x^2+2119x+1980\end{aligned} $$ | |
① | Multiply each term of $ \left( \color{blue}{x+9}\right) $ by each term in $ \left( x+4\right) $. $$ \left( \color{blue}{x+9}\right) \cdot \left( x+4\right) = x^2+4x+9x+36 $$ |
② | Combine like terms: $$ x^2+ \color{blue}{4x} + \color{blue}{9x} +36 = x^2+ \color{blue}{13x} +36 $$ |
③ | Multiply each term of $ \left( \color{blue}{x^2+13x+36}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^2+13x+36}\right) \cdot \left( x+1\right) = x^3+x^2+13x^2+13x+36x+36 $$ |
④ | Combine like terms: $$ x^3+ \color{blue}{x^2} + \color{blue}{13x^2} + \color{red}{13x} + \color{red}{36x} +36 = x^3+ \color{blue}{14x^2} + \color{red}{49x} +36 $$ |
⑤ | Multiply each term of $ \left( \color{blue}{x^3+14x^2+49x+36}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{x^3+14x^2+49x+36}\right) \cdot \left( x-5\right) = x^4-5x^3+14x^3-70x^2+49x^2-245x+36x-180 $$ |
⑥ | Combine like terms: $$ x^4 \color{blue}{-5x^3} + \color{blue}{14x^3} \color{red}{-70x^2} + \color{red}{49x^2} \color{green}{-245x} + \color{green}{36x} -180 = \\ = x^4+ \color{blue}{9x^3} \color{red}{-21x^2} \color{green}{-209x} -180 $$ |
⑦ | Multiply each term of $ \left( \color{blue}{x^4+9x^3-21x^2-209x-180}\right) $ by each term in $ \left( x-11\right) $. $$ \left( \color{blue}{x^4+9x^3-21x^2-209x-180}\right) \cdot \left( x-11\right) = \\ = x^5-11x^4+9x^4-99x^3-21x^3+231x^2-209x^2+2299x-180x+1980 $$ |
⑧ | Combine like terms: $$ x^5 \color{blue}{-11x^4} + \color{blue}{9x^4} \color{red}{-99x^3} \color{red}{-21x^3} + \color{green}{231x^2} \color{green}{-209x^2} + \color{orange}{2299x} \color{orange}{-180x} +1980 = \\ = x^5 \color{blue}{-2x^4} \color{red}{-120x^3} + \color{green}{22x^2} + \color{orange}{2119x} +1980 $$ |