Solve x/x^2+2x+1+1/x^2+5x+4

$$\frac { x } { x ^ { 2 } + 2 x + 1 } + \frac { 1 } { x ^ { 2 } + 5 x + 4 }$$

Evaluate

$$\frac{x^{2}+5x+1}{\left(x+4\right)\left(x+1\right)^{2}}$$

Short Solution Steps

Factor x^{2}+2x+1. Factor x^{2}+5x+4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)^{2} and \left(x+1\right)\left(x+4\right) is \left(x+4\right)\left(x+1\right)^{2}. Multiply \frac{x}{\left(x+1\right)^{2}} times \frac{x+4}{x+4}. Multiply \frac{1}{\left(x+1\right)\left(x+4\right)} times \frac{x+1}{x+1}.

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

Since \frac{x\left(x+4\right)}{\left(x+4\right)\left(x+1\right)^{2}} and \frac{x+1}{\left(x+4\right)\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x\left(x+4\right)+x+1.

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

Combine like terms in x^{2}+4x+x+1.

$$\frac{x^{2}+5x+1}{\left(x+4\right)\left(x+1\right)^{2}}$$

Expand \left(x+4\right)\left(x+1\right)^{2}.

$$\frac{x^{2}+5x+1}{x^{3}+6x^{2}+9x+4}$$

Differentiate w.r.t. x

$$\frac{11-15x-9x^{2}-x^{3}}{\left(x+4\right)^{2}\left(x+1\right)^{3}}$$