Resolver 3beta/alpha+1+3alpha/beta+1

$$\frac { 3 \beta } { \alpha + 1 } + \frac { 3 \alpha } { \beta + 1 }$$

Calcular

$$\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Pasos de solución curta

Para sumar ou restar expresións, expándeas para facer que os seus denominadores sexan iguais. O mínimo común múltiplo de \alpha +1 e \beta +1 é \left(\alpha +1\right)\left(\beta +1\right). Multiplica \frac{3\beta }{\alpha +1} por \frac{\beta +1}{\beta +1}. Multiplica \frac{3\alpha }{\beta +1} por \frac{\alpha +1}{\alpha +1}.

$$\frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}+\frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Dado que \frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} e \frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} teñen o mesmo denominador, súmaos mediante a suma dos seus numeradores.

$$\frac{3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Fai as multiplicacións en 3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right).

$$\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Expande \left(\alpha +1\right)\left(\beta +1\right).

$$\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\alpha \beta +\alpha +\beta +1}$$

Factorizar

$$\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$