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