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Question

Answer

$$x+\dfrac{2}{x^2}+4x+4=\dfrac{5x^3+4x^2+2}{x^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}x+\frac{2}{x^2}+4x+4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x^3+2}{x^2}+4x+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5x^3+2}{x^2}+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5x^3+4x^2+2}{x^2}\end{aligned} $$
①

Add $x$ and $ \dfrac{2}{x^2} $ to get $ \dfrac{ \color{purple}{ x^3+2 } }{ x^2 }$.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x^2 }$.

$$ \begin{aligned} x+ \frac{2}{x^2} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{2}{x^2} = \frac{ x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 2 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^3 } }{ x^2 } + \frac{ \color{purple}{ 2 } }{ x^2 }=\frac{ \color{purple}{ x^3+2 } }{ x^2 } \end{aligned} $$
②

Add $ \dfrac{x^3+2}{x^2} $ and $ 4x $ to get $ \dfrac{ \color{purple}{ 5x^3+2 } }{ x^2 }$.

Step 1: Write $ 4x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^3+2}{x^2} +4x & \xlongequal{\text{Step 1}} \frac{x^3+2}{x^2} + \frac{4x}{\color{red}{1}} = \frac{ x^3+2 }{ x^2 } + \frac{ 4x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^3+2 } }{ x^2 } + \frac{ \color{purple}{ 4x^3 } }{ x^2 }=\frac{ \color{purple}{ 5x^3+2 } }{ x^2 } \end{aligned} $$
③

Add $ \dfrac{5x^3+2}{x^2} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ 5x^3+4x^2+2 } }{ x^2 }$.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{5x^3+2}{x^2} +4 & \xlongequal{\text{Step 1}} \frac{5x^3+2}{x^2} + \frac{4}{\color{red}{1}} = \frac{ 5x^3+2 }{ x^2 } + \frac{ 4 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5x^3+2 } }{ x^2 } + \frac{ \color{purple}{ 4x^2 } }{ x^2 }=\frac{ \color{purple}{ 5x^3+4x^2+2 } }{ x^2 } \end{aligned} $$
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