Solve f(x)=-1/2x(x+4)^2(x+2)(x-1)

$$f ( x ) = - \frac { 1 } { 2 } x ( x + 4 ) ^ { 2 } ( x + 2 ) ( x - 1 )$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$-\frac{1}{2}x\left(x^{2}+8x+16\right)\left(x+2\right)\left(x-1\right)$$

Use the distributive property to multiply -\frac{1}{2}x by x^{2}+8x+16.

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

Use the distributive property to multiply -\frac{1}{2}x^{3}-4x^{2}-8x by x+2 and combine like terms.

$$\left(-\frac{1}{2}x^{4}-5x^{3}-16x^{2}-16x\right)\left(x-1\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{4}-5x^{3}-16x^{2}-16x by x-1 and combine like terms.

$$-\frac{1}{2}x^{5}-\frac{9}{2}x^{4}-11x^{3}+16x$$

Expand

$$-\frac{x^{5}}{2}-\frac{9x^{4}}{2}-11x^{3}+16x$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$-\frac{1}{2}x\left(x^{2}+8x+16\right)\left(x+2\right)\left(x-1\right)$$

Use the distributive property to multiply -\frac{1}{2}x by x^{2}+8x+16.

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

Use the distributive property to multiply -\frac{1}{2}x^{3}-4x^{2}-8x by x+2 and combine like terms.

$$\left(-\frac{1}{2}x^{4}-5x^{3}-16x^{2}-16x\right)\left(x-1\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{4}-5x^{3}-16x^{2}-16x by x-1 and combine like terms.

$$-\frac{1}{2}x^{5}-\frac{9}{2}x^{4}-11x^{3}+16x$$