Solve x+1/4x^2-4x+1+3/2x^2+5x-3

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

Evaluate

$$\frac{x\left(x+10\right)}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Short Solution Steps

Factor 4x^{2}-4x+1. Factor 2x^{2}+5x-3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)^{2} and \left(2x-1\right)\left(x+3\right) is \left(x+3\right)\left(2x-1\right)^{2}. Multiply \frac{x+1}{\left(2x-1\right)^{2}} times \frac{x+3}{x+3}. Multiply \frac{3}{\left(2x-1\right)\left(x+3\right)} times \frac{2x-1}{2x-1}.

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

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

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

Do the multiplications in \left(x+1\right)\left(x+3\right)+3\left(2x-1\right).

$$\frac{x^{2}+3x+x+3+6x-3}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Combine like terms in x^{2}+3x+x+3+6x-3.

$$\frac{x^{2}+10x}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Expand \left(x+3\right)\left(2x-1\right)^{2}.

$$\frac{x^{2}+10x}{4x^{3}+8x^{2}-11x+3}$$

Expand

$$\frac{x^{2}+10x}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Short Solution Steps

Factor 4x^{2}-4x+1. Factor 2x^{2}+5x-3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)^{2} and \left(2x-1\right)\left(x+3\right) is \left(x+3\right)\left(2x-1\right)^{2}. Multiply \frac{x+1}{\left(2x-1\right)^{2}} times \frac{x+3}{x+3}. Multiply \frac{3}{\left(2x-1\right)\left(x+3\right)} times \frac{2x-1}{2x-1}.

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

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

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

Do the multiplications in \left(x+1\right)\left(x+3\right)+3\left(2x-1\right).

$$\frac{x^{2}+3x+x+3+6x-3}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Combine like terms in x^{2}+3x+x+3+6x-3.

$$\frac{x^{2}+10x}{\left(x+3\right)\left(2x-1\right)^{2}}$$

Expand \left(x+3\right)\left(2x-1\right)^{2}.

$$\frac{x^{2}+10x}{4x^{3}+8x^{2}-11x+3}$$