Answer
$$(x+3)^2(x-3)^2+4=x^4-18x^2+85$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+3)^2(x-3)^2+4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+6x+9)(x^2-6x+9)+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^4-18x^2+81+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4-18x^2+85\end{aligned} $$ | |
| ① | Find $ \left(x+3\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}{ 3 }$. $$ \begin{aligned}\left(x+3\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 3 + \color{red}{3^2} = x^2+6x+9\end{aligned} $$Find $ \left(x-3\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}{ 3 }$. $$ \begin{aligned}\left(x-3\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 3 + \color{red}{3^2} = x^2-6x+9\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+6x+9}\right) $ by each term in $ \left( x^2-6x+9\right) $. $$ \left( \color{blue}{x^2+6x+9}\right) \cdot \left( x^2-6x+9\right) = \\ = x^4 -\cancel{6x^3}+9x^2+ \cancel{6x^3}-36x^2+ \cancel{54x}+9x^2 -\cancel{54x}+81 $$ |
| ③ | Combine like terms: $$ x^4 \, \color{blue}{ -\cancel{6x^3}} \,+ \color{green}{9x^2} + \, \color{blue}{ \cancel{6x^3}} \, \color{orange}{-36x^2} + \, \color{blue}{ \cancel{54x}} \,+ \color{orange}{9x^2} \, \color{blue}{ -\cancel{54x}} \,+81 = x^4 \color{orange}{-18x^2} +81 $$ |
| ④ | Combine like terms: $$ x^4-18x^2+ \color{blue}{81} + \color{blue}{4} = x^4-18x^2+ \color{blue}{85} $$ |
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