Evaluate
\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}
Factor
\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}
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\frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}+\frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \alpha +1 and \beta +1 is \left(\alpha +1\right)\left(\beta +1\right). Multiply \frac{3\beta }{\alpha +1} times \frac{\beta +1}{\beta +1}. Multiply \frac{3\alpha }{\beta +1} times \frac{\alpha +1}{\alpha +1}.
\frac{3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}
Since \frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} and \frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} have the same denominator, add them by adding their numerators.
\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\left(\alpha +1\right)\left(\beta +1\right)}
Do the multiplications in 3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right).
\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\alpha \beta +\alpha +\beta +1}
Expand \left(\alpha +1\right)\left(\beta +1\right).
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
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}