Solve x^2+2x+3+21/x+1/x^2

$$x ^ { 2 } + 2 x + 3 + 2 \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

$$x^{2}+2x+3+\frac{2}{x}+\frac{1}{x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+2x+3 times \frac{x}{x}.

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

Since \frac{\left(x^{2}+2x+3\right)x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+2x+3\right)x+2.

$$\frac{x^{3}+2x^{2}+3x+2}{x}+\frac{1}{x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{x^{3}+2x^{2}+3x+2}{x} times \frac{x}{x}.

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

Since \frac{\left(x^{3}+2x^{2}+3x+2\right)x}{x^{2}} and \frac{1}{x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{3}+2x^{2}+3x+2\right)x+1.

$$\frac{x^{4}+2x^{3}+3x^{2}+2x+1}{x^{2}}$$

Factor

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