Question
Answer
$$(x+1)^2(x+2)^2(2(x+1)^2+2(x+1)-1)=2x^6+18x^5+65x^4+120x^3+119x^2+60x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)^2(x+2)^2(2(x+1)^2+2(x+1)-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+2x+1)(x^2+4x+4)(2(x^2+2x+1)+2(x+1)-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^4+6x^3+13x^2+12x+4)(2x^2+4x+2+2x+2-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^4+6x^3+13x^2+12x+4)(2x^2+6x+4-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^4+6x^3+13x^2+12x+4)(2x^2+6x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}2x^6+18x^5+65x^4+120x^3+119x^2+60x+12\end{aligned} $$ | |
| ① | Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$Find $ \left(x+2\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(x+2\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 2 + \color{red}{2^2} = x^2+4x+4\end{aligned} $$Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+2x+1}\right) $ by each term in $ \left( x^2+4x+4\right) $. $$ \left( \color{blue}{x^2+2x+1}\right) \cdot \left( x^2+4x+4\right) = x^4+4x^3+4x^2+2x^3+8x^2+8x+x^2+4x+4 $$ |
| ③ | Combine like terms: $$ x^4+ \color{blue}{4x^3} + \color{red}{4x^2} + \color{blue}{2x^3} + \color{green}{8x^2} + \color{orange}{8x} + \color{green}{x^2} + \color{orange}{4x} +4 = \\ = x^4+ \color{blue}{6x^3} + \color{green}{13x^2} + \color{orange}{12x} +4 $$Multiply $ \color{blue}{2} $ by $ \left( x^2+2x+1\right) $ $$ \color{blue}{2} \cdot \left( x^2+2x+1\right) = 2x^2+4x+2 $$Multiply $ \color{blue}{2} $ by $ \left( x+1\right) $ $$ \color{blue}{2} \cdot \left( x+1\right) = 2x+2 $$ |
| ④ | Combine like terms: $$ 2x^2+ \color{blue}{4x} + \color{red}{2} + \color{blue}{2x} + \color{red}{2} = 2x^2+ \color{blue}{6x} + \color{red}{4} $$ |
| ⑤ | Combine like terms: $$ 2x^2+6x+ \color{blue}{4} \color{blue}{-1} = 2x^2+6x+ \color{blue}{3} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^4+6x^3+13x^2+12x+4}\right) $ by each term in $ \left( 2x^2+6x+3\right) $. $$ \left( \color{blue}{x^4+6x^3+13x^2+12x+4}\right) \cdot \left( 2x^2+6x+3\right) = \\ = 2x^6+6x^5+3x^4+12x^5+36x^4+18x^3+26x^4+78x^3+39x^2+24x^3+72x^2+36x+8x^2+24x+12 $$ |
| ⑦ | Combine like terms: $$ 2x^6+ \color{blue}{6x^5} + \color{red}{3x^4} + \color{blue}{12x^5} + \color{green}{36x^4} + \color{orange}{18x^3} + \color{green}{26x^4} + \color{blue}{78x^3} + \color{red}{39x^2} + \color{blue}{24x^3} + \color{green}{72x^2} + \color{orange}{36x} + \color{green}{8x^2} + \color{orange}{24x} +12 = \\ = 2x^6+ \color{blue}{18x^5} + \color{green}{65x^4} + \color{blue}{120x^3} + \color{green}{119x^2} + \color{orange}{60x} +12 $$ |
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