Solve =2/p*2/p-1/1+frac{2{p-1}*1}*2/4p^2+4p+1

$$= \frac { 2 } { p } \cdot \frac { \frac { 2 } { p - 1 } } { 1 + \frac { 2 } { p - 1 } \cdot 1 } \cdot \frac { 2 } { 4 p ^ { 2 } + 4 p + 1 }$$

Evaluate

$$\frac{8}{p\left(p+1\right)\left(2p+1\right)^{2}}$$

Short Solution Steps

Express \frac{2}{p-1}\times 1 as a single fraction.

$$\frac{2}{p}\times \frac{\frac{2}{p-1}}{1+\frac{2}{p-1}}\times \frac{2}{4p^{2}+4p+1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{p-1}{p-1}.

$$\frac{2}{p}\times \frac{\frac{2}{p-1}}{\frac{p-1}{p-1}+\frac{2}{p-1}}\times \frac{2}{4p^{2}+4p+1}$$

Since \frac{p-1}{p-1} and \frac{2}{p-1} have the same denominator, add them by adding their numerators.

$$\frac{2}{p}\times \frac{\frac{2}{p-1}}{\frac{p-1+2}{p-1}}\times \frac{2}{4p^{2}+4p+1}$$

Combine like terms in p-1+2.

$$\frac{2}{p}\times \frac{\frac{2}{p-1}}{\frac{p+1}{p-1}}\times \frac{2}{4p^{2}+4p+1}$$

Divide \frac{2}{p-1} by \frac{p+1}{p-1} by multiplying \frac{2}{p-1} by the reciprocal of \frac{p+1}{p-1}.

$$\frac{2}{p}\times \frac{2\left(p-1\right)}{\left(p-1\right)\left(p+1\right)}\times \frac{2}{4p^{2}+4p+1}$$

Cancel out p-1 in both numerator and denominator.

$$\frac{2}{p}\times \frac{2}{p+1}\times \frac{2}{4p^{2}+4p+1}$$

Multiply \frac{2}{p} times \frac{2}{p+1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 2}{p\left(p+1\right)}\times \frac{2}{4p^{2}+4p+1}$$

Multiply \frac{2\times 2}{p\left(p+1\right)} times \frac{2}{4p^{2}+4p+1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 2\times 2}{p\left(p+1\right)\left(4p^{2}+4p+1\right)}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 2}{p\left(p+1\right)\left(4p^{2}+4p+1\right)}$$

Multiply 4 and 2 to get 8.

$$\frac{8}{p\left(p+1\right)\left(4p^{2}+4p+1\right)}$$

Use the distributive property to multiply p by p+1.

$$\frac{8}{\left(p^{2}+p\right)\left(4p^{2}+4p+1\right)}$$

Use the distributive property to multiply p^{2}+p by 4p^{2}+4p+1 and combine like terms.

$$\frac{8}{4p^{4}+8p^{3}+5p^{2}+p}$$

Factor

$$\frac{8}{p\left(p+1\right)\left(2p+1\right)^{2}}$$