Solve x+x^2/5(16+8x+2x^2/2+x)^2-40

$$\frac { x + x ^ { 2 } } { 5 } ( \frac { 16 + 8 x + 2 x ^ { 2 } } { 2 + x } ) ^ { 2 } - 40$$

Evaluate

$$\frac{4\left(x^{6}+9x^{5}+40x^{4}+96x^{3}+78x^{2}-136x-200\right)}{5\left(x+2\right)^{2}}$$

Short Solution Steps

To raise \frac{16+8x+2x^{2}}{2+x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{x+x^{2}}{5}\times \frac{\left(16+8x+2x^{2}\right)^{2}}{\left(2+x\right)^{2}}-40$$

Multiply \frac{x+x^{2}}{5} times \frac{\left(16+8x+2x^{2}\right)^{2}}{\left(2+x\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}}-40$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 40 times \frac{5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}}-\frac{40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}$$

Since \frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}} and \frac{40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}-40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}$$

Do the multiplications in \left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}-40\times 5\left(2+x\right)^{2}.

$$\frac{256x+256x^{2}+128x^{3}+32x^{4}+4x^{5}+256x^{2}+256x^{3}+128x^{4}+32x^{5}+4x^{6}-800-800x-200x^{2}}{5\left(2+x\right)^{2}}$$

Combine like terms in 256x+256x^{2}+128x^{3}+32x^{4}+4x^{5}+256x^{2}+256x^{3}+128x^{4}+32x^{5}+4x^{6}-800-800x-200x^{2}.

$$\frac{4x^{6}-544x+312x^{2}+384x^{3}+160x^{4}+36x^{5}-800}{5\left(2+x\right)^{2}}$$

Expand 5\left(2+x\right)^{2}.

$$\frac{4x^{6}-544x+312x^{2}+384x^{3}+160x^{4}+36x^{5}-800}{5x^{2}+20x+20}$$

Expand

$$\frac{4\left(x^{6}+9x^{5}+40x^{4}+96x^{3}+78x^{2}-136x-200\right)}{5\left(x+2\right)^{2}}$$

Short Solution Steps

To raise \frac{16+8x+2x^{2}}{2+x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{x+x^{2}}{5}\times \frac{\left(16+8x+2x^{2}\right)^{2}}{\left(2+x\right)^{2}}-40$$

Multiply \frac{x+x^{2}}{5} times \frac{\left(16+8x+2x^{2}\right)^{2}}{\left(2+x\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}}-40$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 40 times \frac{5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}}-\frac{40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}$$

Since \frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}}{5\left(2+x\right)^{2}} and \frac{40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}-40\times 5\left(2+x\right)^{2}}{5\left(2+x\right)^{2}}$$

Do the multiplications in \left(x+x^{2}\right)\left(16+8x+2x^{2}\right)^{2}-40\times 5\left(2+x\right)^{2}.

$$\frac{256x+256x^{2}+128x^{3}+32x^{4}+4x^{5}+256x^{2}+256x^{3}+128x^{4}+32x^{5}+4x^{6}-800-800x-200x^{2}}{5\left(2+x\right)^{2}}$$

Combine like terms in 256x+256x^{2}+128x^{3}+32x^{4}+4x^{5}+256x^{2}+256x^{3}+128x^{4}+32x^{5}+4x^{6}-800-800x-200x^{2}.

$$\frac{4x^{6}-544x+312x^{2}+384x^{3}+160x^{4}+36x^{5}-800}{5\left(2+x\right)^{2}}$$

Expand 5\left(2+x\right)^{2}.

$$\frac{4x^{6}-544x+312x^{2}+384x^{3}+160x^{4}+36x^{5}-800}{5x^{2}+20x+20}$$