Question
Answer
$$(11x+2)^4-2(x-1)(x+3)(3x^3+5)=-6x^5+14629x^4+10666x^3+2894x^2+332x+46$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(11x+2)^4-2(x-1)(x+3)(3x^3+5)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}14641x^4+10648x^3+2904x^2+352x+16-2(x-1)(x+3)(3x^3+5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}14641x^4+10648x^3+2904x^2+352x+16-(2x-2)(x+3)(3x^3+5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}14641x^4+10648x^3+2904x^2+352x+16-(2x^2+6x-2x-6)(3x^3+5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}14641x^4+10648x^3+2904x^2+352x+16-(2x^2+4x-6)(3x^3+5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}14641x^4+10648x^3+2904x^2+352x+16-(6x^5+10x^2+12x^4+20x-18x^3-30) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}14641x^4+10648x^3+2904x^2+352x+16-6x^5-10x^2-12x^4-20x+18x^3+30 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}-6x^5+14629x^4+10666x^3+2894x^2+332x+46\end{aligned} $$ | |
| ① | $$ (11x+2)^4 = (11x+2)^2 \cdot (11x+2)^2 $$ |
| ② | Find $ \left(11x+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}{ 11x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(11x+2\right)^2 = \color{blue}{\left( 11x \right)^2} +2 \cdot 11x \cdot 2 + \color{red}{2^2} = 121x^2+44x+4\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{121x^2+44x+4}\right) $ by each term in $ \left( 121x^2+44x+4\right) $. $$ \left( \color{blue}{121x^2+44x+4}\right) \cdot \left( 121x^2+44x+4\right) = \\ = 14641x^4+5324x^3+484x^2+5324x^3+1936x^2+176x+484x^2+176x+16 $$ |
| ④ | Combine like terms: $$ 14641x^4+ \color{blue}{5324x^3} + \color{red}{484x^2} + \color{blue}{5324x^3} + \color{green}{1936x^2} + \color{orange}{176x} + \color{green}{484x^2} + \color{orange}{176x} +16 = \\ = 14641x^4+ \color{blue}{10648x^3} + \color{green}{2904x^2} + \color{orange}{352x} +16 $$ |
| ⑤ | Multiply $ \color{blue}{2} $ by $ \left( x-1\right) $ $$ \color{blue}{2} \cdot \left( x-1\right) = 2x-2 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{2x-2}\right) $ by each term in $ \left( x+3\right) $. $$ \left( \color{blue}{2x-2}\right) \cdot \left( x+3\right) = 2x^2+6x-2x-6 $$ |
| ⑦ | Combine like terms: $$ 2x^2+ \color{blue}{6x} \color{blue}{-2x} -6 = 2x^2+ \color{blue}{4x} -6 $$ |
| ⑧ | Multiply each term of $ \left( \color{blue}{2x^2+4x-6}\right) $ by each term in $ \left( 3x^3+5\right) $. $$ \left( \color{blue}{2x^2+4x-6}\right) \cdot \left( 3x^3+5\right) = 6x^5+10x^2+12x^4+20x-18x^3-30 $$ |
| ⑨ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 6x^5+10x^2+12x^4+20x-18x^3-30 \right) = -6x^5-10x^2-12x^4-20x+18x^3+30 $$ |
| ⑩ | Combine like terms: $$ \color{blue}{14641x^4} + \color{red}{10648x^3} + \color{green}{2904x^2} + \color{orange}{352x} + \color{blue}{16} -6x^5 \color{green}{-10x^2} \color{blue}{-12x^4} \color{orange}{-20x} + \color{red}{18x^3} + \color{blue}{30} = \\ = -6x^5+ \color{blue}{14629x^4} + \color{red}{10666x^3} + \color{green}{2894x^2} + \color{orange}{332x} + \color{blue}{46} $$ |
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