Evaluate
\frac{\alpha ^{2}+4\alpha +\beta ^{2}+4\beta +8}{\left(\alpha +2\right)\left(\beta +2\right)}
Expand
\frac{\alpha ^{2}+4\alpha +\beta ^{2}+4\beta +8}{\left(\alpha +2\right)\left(\beta +2\right)}
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\frac{\left(\alpha +2\right)\left(\alpha +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}+\frac{\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \beta +2 and \alpha +2 is \left(\alpha +2\right)\left(\beta +2\right). Multiply \frac{\alpha +2}{\beta +2} times \frac{\alpha +2}{\alpha +2}. Multiply \frac{\beta +2}{\alpha +2} times \frac{\beta +2}{\beta +2}.
\frac{\left(\alpha +2\right)\left(\alpha +2\right)+\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}
Since \frac{\left(\alpha +2\right)\left(\alpha +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)} and \frac{\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)} have the same denominator, add them by adding their numerators.
\frac{\alpha ^{2}+2\alpha +2\alpha +4+\beta ^{2}+2\beta +2\beta +4}{\left(\alpha +2\right)\left(\beta +2\right)}
Do the multiplications in \left(\alpha +2\right)\left(\alpha +2\right)+\left(\beta +2\right)\left(\beta +2\right).
\frac{\alpha ^{2}+4\alpha +8+\beta ^{2}+4\beta }{\left(\alpha +2\right)\left(\beta +2\right)}
Combine like terms in \alpha ^{2}+2\alpha +2\alpha +4+\beta ^{2}+2\beta +2\beta +4.
\frac{\alpha ^{2}+4\alpha +8+\beta ^{2}+4\beta }{\alpha \beta +2\alpha +2\beta +4}
Expand \left(\alpha +2\right)\left(\beta +2\right).
\frac{\left(\alpha +2\right)\left(\alpha +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}+\frac{\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \beta +2 and \alpha +2 is \left(\alpha +2\right)\left(\beta +2\right). Multiply \frac{\alpha +2}{\beta +2} times \frac{\alpha +2}{\alpha +2}. Multiply \frac{\beta +2}{\alpha +2} times \frac{\beta +2}{\beta +2}.
\frac{\left(\alpha +2\right)\left(\alpha +2\right)+\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)}
Since \frac{\left(\alpha +2\right)\left(\alpha +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)} and \frac{\left(\beta +2\right)\left(\beta +2\right)}{\left(\alpha +2\right)\left(\beta +2\right)} have the same denominator, add them by adding their numerators.
\frac{\alpha ^{2}+2\alpha +2\alpha +4+\beta ^{2}+2\beta +2\beta +4}{\left(\alpha +2\right)\left(\beta +2\right)}
Do the multiplications in \left(\alpha +2\right)\left(\alpha +2\right)+\left(\beta +2\right)\left(\beta +2\right).
\frac{\alpha ^{2}+4\alpha +8+\beta ^{2}+4\beta }{\left(\alpha +2\right)\left(\beta +2\right)}
Combine like terms in \alpha ^{2}+2\alpha +2\alpha +4+\beta ^{2}+2\beta +2\beta +4.
\frac{\alpha ^{2}+4\alpha +8+\beta ^{2}+4\beta }{\alpha \beta +2\alpha +2\beta +4}
Expand \left(\alpha +2\right)\left(\beta +2\right).
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}