Answer
$$(1-x)(6-x)(5-x)(3-x-4)+2((3-x)\cdot(-2)+8)=x^4-11x^3+29x^2+15x-26$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-x)(6-x)(5-x)(3-x-4)+2((3-x)\cdot(-2)+8)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1-x)(6-x)(5-x)(-x-1)+2((3-x)\cdot(-2)+8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1-x)(6-x)(-5x-5+x^2+x)+2(-6+2x+8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1-x)(6-x)(x^2-4x-5)+2(2x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1-x)(6x^2-24x-30-x^3+4x^2+5x)+4x+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(1-x)(-x^3+10x^2-19x-30)+4x+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^4-11x^3+29x^2+11x-30+4x+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^4-11x^3+29x^2+15x-26\end{aligned} $$ | |
| ① | Combine like terms: $$ \color{blue}{3} -x \color{blue}{-4} = -x \color{blue}{-1} $$ |
| ② | Multiply each term of $ \left( \color{blue}{5-x}\right) $ by each term in $ \left( -x-1\right) $. $$ \left( \color{blue}{5-x}\right) \cdot \left( -x-1\right) = -5x-5+x^2+x $$$$ \left( \color{blue}{3-x}\right) \cdot -2 = -6+2x $$ |
| ③ | Combine like terms: $$ \color{blue}{-5x} -5+x^2+ \color{blue}{x} = x^2 \color{blue}{-4x} -5 $$Combine like terms: $$ \color{blue}{-6} +2x+ \color{blue}{8} = 2x+ \color{blue}{2} $$ |
| ④ | Multiply each term of $ \left( \color{blue}{6-x}\right) $ by each term in $ \left( x^2-4x-5\right) $. $$ \left( \color{blue}{6-x}\right) \cdot \left( x^2-4x-5\right) = 6x^2-24x-30-x^3+4x^2+5x $$Multiply $ \color{blue}{2} $ by $ \left( 2x+2\right) $ $$ \color{blue}{2} \cdot \left( 2x+2\right) = 4x+4 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{6x^2} \color{red}{-24x} -30-x^3+ \color{blue}{4x^2} + \color{red}{5x} = -x^3+ \color{blue}{10x^2} \color{red}{-19x} -30 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{1-x}\right) $ by each term in $ \left( -x^3+10x^2-19x-30\right) $. $$ \left( \color{blue}{1-x}\right) \cdot \left( -x^3+10x^2-19x-30\right) = -x^3+10x^2-19x-30+x^4-10x^3+19x^2+30x $$ |
| ⑦ | Combine like terms: $$ \color{blue}{-x^3} + \color{red}{10x^2} \color{green}{-19x} -30+x^4 \color{blue}{-10x^3} + \color{red}{19x^2} + \color{green}{30x} = x^4 \color{blue}{-11x^3} + \color{red}{29x^2} + \color{green}{11x} -30 $$ |
| ⑧ | Combine like terms: $$ x^4-11x^3+29x^2+ \color{blue}{11x} \color{red}{-30} + \color{blue}{4x} + \color{red}{4} = x^4-11x^3+29x^2+ \color{blue}{15x} \color{red}{-26} $$ |
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