Question
Answer
$$(-2x^2+8x+3)^2-(-x^2+4x+3)^2=3x^4-24x^3+42x^2+24x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-2x^2+8x+3)^2-(-x^2+4x+3)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4x^4-32x^3+52x^2+48x+9-(x^4-8x^3+10x^2+24x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4x^4-32x^3+52x^2+48x+9-x^4+8x^3-10x^2-24x-9 \xlongequal{ } \\[1 em] & \xlongequal{ }4x^4-32x^3+52x^2+48x+ \cancel{9}-x^4+8x^3-10x^2-24x -\cancel{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3x^4-24x^3+42x^2+24x\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{-2x^2+8x+3}\right) $ by each term in $ \left( -2x^2+8x+3\right) $. $$ \left( \color{blue}{-2x^2+8x+3}\right) \cdot \left( -2x^2+8x+3\right) = 4x^4-16x^3-6x^2-16x^3+64x^2+24x-6x^2+24x+9 $$ |
| ② | Combine like terms: $$ 4x^4 \color{blue}{-16x^3} \color{red}{-6x^2} \color{blue}{-16x^3} + \color{green}{64x^2} + \color{orange}{24x} \color{green}{-6x^2} + \color{orange}{24x} +9 = \\ = 4x^4 \color{blue}{-32x^3} + \color{green}{52x^2} + \color{orange}{48x} +9 $$Multiply each term of $ \left( \color{blue}{-x^2+4x+3}\right) $ by each term in $ \left( -x^2+4x+3\right) $. $$ \left( \color{blue}{-x^2+4x+3}\right) \cdot \left( -x^2+4x+3\right) = x^4-4x^3-3x^2-4x^3+16x^2+12x-3x^2+12x+9 $$ |
| ③ | Combine like terms: $$ x^4 \color{blue}{-4x^3} \color{red}{-3x^2} \color{blue}{-4x^3} + \color{green}{16x^2} + \color{orange}{12x} \color{green}{-3x^2} + \color{orange}{12x} +9 = \\ = x^4 \color{blue}{-8x^3} + \color{green}{10x^2} + \color{orange}{24x} +9 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( x^4-8x^3+10x^2+24x+9 \right) = -x^4+8x^3-10x^2-24x-9 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{4x^4} \color{red}{-32x^3} + \color{green}{52x^2} + \color{orange}{48x} + \, \color{blue}{ \cancel{9}} \, \color{blue}{-x^4} + \color{red}{8x^3} \color{green}{-10x^2} \color{orange}{-24x} \, \color{blue}{ -\cancel{9}} \, = \\ = \color{blue}{3x^4} \color{red}{-24x^3} + \color{green}{42x^2} + \color{orange}{24x} $$ |
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