Solve [(a^5-1):(a-1)-a^4]:(a+1)+(a-1)(a

$$[ ( a ^ { 5 } - 1 ) : ( a - 1 ) - a ^ { 4 } ] : ( a + 1 ) + ( a - 1 ) ( a$$

Evaluate

$$a\left(a-1\right)+a^{2}+1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{5}-1}{a-1}.

$$\frac{\frac{\left(a-1\right)\left(a^{4}+a^{3}+a^{2}+a+1\right)}{a-1}-a^{4}}{a+1}+\left(a-1\right)a$$

Cancel out a-1 in both numerator and denominator.

$$\frac{a^{4}+a^{3}+a^{2}+a+1-a^{4}}{a+1}+\left(a-1\right)a$$

Combine a^{4} and -a^{4} to get 0.

$$\frac{a^{3}+a^{2}+a+1}{a+1}+\left(a-1\right)a$$

Factor the expressions that are not already factored in \frac{a^{3}+a^{2}+a+1}{a+1}.

$$\frac{\left(a+1\right)\left(a^{2}+1\right)}{a+1}+\left(a-1\right)a$$

Cancel out a+1 in both numerator and denominator.

$$a^{2}+1+\left(a-1\right)a$$

Use the distributive property to multiply a-1 by a.

$$a^{2}+1+a^{2}-a$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}+1-a$$

Expand

$$2a^{2}-a+1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{5}-1}{a-1}.

$$\frac{\frac{\left(a-1\right)\left(a^{4}+a^{3}+a^{2}+a+1\right)}{a-1}-a^{4}}{a+1}+\left(a-1\right)a$$

Cancel out a-1 in both numerator and denominator.

$$\frac{a^{4}+a^{3}+a^{2}+a+1-a^{4}}{a+1}+\left(a-1\right)a$$

Combine a^{4} and -a^{4} to get 0.

$$\frac{a^{3}+a^{2}+a+1}{a+1}+\left(a-1\right)a$$

Factor the expressions that are not already factored in \frac{a^{3}+a^{2}+a+1}{a+1}.

$$\frac{\left(a+1\right)\left(a^{2}+1\right)}{a+1}+\left(a-1\right)a$$

Cancel out a+1 in both numerator and denominator.

$$a^{2}+1+\left(a-1\right)a$$

Use the distributive property to multiply a-1 by a.

$$a^{2}+1+a^{2}-a$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}+1-a$$