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

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

Evaluate

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

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)^{2} and \left(2x-1\right)\left(x+2\right) is \left(2x-1\right)\left(x+2\right)^{2}. Multiply \frac{4}{\left(x+2\right)^{2}} times \frac{2x-1}{2x-1}. Multiply \frac{3}{\left(2x-1\right)\left(x+2\right)} times \frac{x+2}{x+2}.

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

Since \frac{4\left(2x-1\right)}{\left(2x-1\right)\left(x+2\right)^{2}} and \frac{3\left(x+2\right)}{\left(2x-1\right)\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(2x-1\right)-3\left(x+2\right).

$$\frac{8x-4-3x-6}{\left(2x-1\right)\left(x+2\right)^{2}}$$

Combine like terms in 8x-4-3x-6.

$$\frac{5x-10}{\left(2x-1\right)\left(x+2\right)^{2}}$$

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

$$\frac{5x-10}{2x^{3}+7x^{2}+4x-4}$$

Differentiate w.r.t. x

$$\frac{5\left(2+13x-4x^{2}\right)}{4x^{5}+20x^{4}+25x^{3}-10x^{2}-20x+8}$$