Solve -1/2hleft(h+1right)left(2h+1right)+hleft(n+1right)

$$- \frac{ 1 }{ 2 } h \left( h+1 \right) \left( 2h+1 \right) +h \left( n+1 \right)$$

Evaluate

$$hn-h^{3}-\frac{3h^{2}}{2}+\frac{h}{2}$$

Solution Steps

Use the distributive property to multiply -\frac{1}{2}h by h+1.

$$\left(-\frac{1}{2}hh-\frac{1}{2}h\right)\left(2h+1\right)+h\left(n+1\right)$$

Multiply h and h to get h^{2}.

$$\left(-\frac{1}{2}h^{2}-\frac{1}{2}h\right)\left(2h+1\right)+h\left(n+1\right)$$

Apply the distributive property by multiplying each term of -\frac{1}{2}h^{2}-\frac{1}{2}h by each term of 2h+1.

$$-\frac{1}{2}h^{2}\times 2h-\frac{1}{2}h^{2}-\frac{1}{2}h\times 2h-\frac{1}{2}h+h\left(n+1\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$-\frac{1}{2}h^{3}\times 2-\frac{1}{2}h^{2}-\frac{1}{2}h\times 2h-\frac{1}{2}h+h\left(n+1\right)$$

Multiply h and h to get h^{2}.

$$-\frac{1}{2}h^{3}\times 2-\frac{1}{2}h^{2}-\frac{1}{2}h^{2}\times 2-\frac{1}{2}h+h\left(n+1\right)$$

Cancel out 2 and 2.

$$-h^{3}-\frac{1}{2}h^{2}-\frac{1}{2}h^{2}\times 2-\frac{1}{2}h+h\left(n+1\right)$$

Cancel out 2 and 2.

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

Combine -\frac{1}{2}h^{2} and -h^{2} to get -\frac{3}{2}h^{2}.

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

Use the distributive property to multiply h by n+1.

$$-h^{3}-\frac{3}{2}h^{2}-\frac{1}{2}h+hn+h$$

Combine -\frac{1}{2}h and h to get \frac{1}{2}h.

$$-h^{3}-\frac{3}{2}h^{2}+\frac{1}{2}h+hn$$

Expand

$$hn-h^{3}-\frac{3h^{2}}{2}+\frac{h}{2}$$

Solution Steps

Use the distributive property to multiply -\frac{1}{2}h by h+1.

$$\left(-\frac{1}{2}hh-\frac{1}{2}h\right)\left(2h+1\right)+h\left(n+1\right)$$

Multiply h and h to get h^{2}.

$$\left(-\frac{1}{2}h^{2}-\frac{1}{2}h\right)\left(2h+1\right)+h\left(n+1\right)$$

Apply the distributive property by multiplying each term of -\frac{1}{2}h^{2}-\frac{1}{2}h by each term of 2h+1.

$$-\frac{1}{2}h^{2}\times 2h-\frac{1}{2}h^{2}-\frac{1}{2}h\times 2h-\frac{1}{2}h+h\left(n+1\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$-\frac{1}{2}h^{3}\times 2-\frac{1}{2}h^{2}-\frac{1}{2}h\times 2h-\frac{1}{2}h+h\left(n+1\right)$$

Multiply h and h to get h^{2}.

$$-\frac{1}{2}h^{3}\times 2-\frac{1}{2}h^{2}-\frac{1}{2}h^{2}\times 2-\frac{1}{2}h+h\left(n+1\right)$$

Cancel out 2 and 2.

$$-h^{3}-\frac{1}{2}h^{2}-\frac{1}{2}h^{2}\times 2-\frac{1}{2}h+h\left(n+1\right)$$

Cancel out 2 and 2.

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

Combine -\frac{1}{2}h^{2} and -h^{2} to get -\frac{3}{2}h^{2}.

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

Use the distributive property to multiply h by n+1.

$$-h^{3}-\frac{3}{2}h^{2}-\frac{1}{2}h+hn+h$$

Combine -\frac{1}{2}h and h to get \frac{1}{2}h.

$$-h^{3}-\frac{3}{2}h^{2}+\frac{1}{2}h+hn$$