Answer
$$0.5x(x+1)(x+1)(x+2)+(x+1)(x+2)(2x+3)=2x^3+9x^2+13x+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}0.5x(x+1)(x+1)(x+2)+(x+1)(x+2)(2x+3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}0x(x+1)(x+1)(x+2)+(x^2+2x+x+2)(2x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}0x(x+1)(x+1)(x+2)+(x^2+3x+2)(2x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(0x^2+0x)(x+1)(x+2)+2x^3+3x^2+6x^2+9x+4x+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(0x^2+0x)(x+1)(x+2)+2x^3+9x^2+13x+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(0x^3+0x^2+0x^2+0x)(x+2)+2x^3+9x^2+13x+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}0(x+2)+2x^3+9x^2+13x+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}0x+0+2x^3+9x^2+13x+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}2x^3+9x^2+13x+6\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x+1}\right) $ by each term in $ \left( x+2\right) $. $$ \left( \color{blue}{x+1}\right) \cdot \left( x+2\right) = x^2+2x+x+2 $$ |
| ② | Combine like terms: $$ x^2+ \color{blue}{2x} + \color{blue}{x} +2 = x^2+ \color{blue}{3x} +2 $$ |
| ③ | Multiply $ \color{blue}{0x} $ by $ \left( x+1\right) $ $$ \color{blue}{0x} \cdot \left( x+1\right) = 0x^20x $$ Multiply each term of $ \left( \color{blue}{x^2+3x+2}\right) $ by each term in $ \left( 2x+3\right) $. $$ \left( \color{blue}{x^2+3x+2}\right) \cdot \left( 2x+3\right) = 2x^3+3x^2+6x^2+9x+4x+6 $$ |
| ④ | Combine like terms: $$ 2x^3+ \color{blue}{3x^2} + \color{blue}{6x^2} + \color{red}{9x} + \color{red}{4x} +6 = 2x^3+ \color{blue}{9x^2} + \color{red}{13x} +6 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{0x^20x}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{0x^20x}\right) \cdot \left( x+1\right) = 0x^3 \cancel{0x^2} \cancel{0x^2}0x $$ |
| ⑥ | Combine like terms: $$ 0x^3 \, \color{blue}{ \cancel{0x^2}} \, \, \color{blue}{ \cancel{0x^2}} \,0x = 0 $$ |
| ⑦ | Multiply $ \color{blue}{0} $ by $ \left( x+2\right) $ $$ \color{blue}{0} \cdot \left( x+2\right) = 0x0 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{0x} \color{red}{0} +2x^3+9x^2+ \color{blue}{13x} + \color{red}{6} = 2x^3+9x^2+ \color{blue}{13x} + \color{red}{6} $$ |
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