Solve =(k+1)(k+2)(k+3)(k+4)/4=R.H.S

$$= \frac { ( k + 1 ) ( k + 2 ) ( k + 3 ) ( k + 4 ) } { 4 } = R . H . S$$

Solve for R

$$R=\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)\left(k+4\right)}{4}$$

Solution Steps

Use the distributive property to multiply k+1 by k+2 and combine like terms.

$$\frac{\left(k^{2}+3k+2\right)\left(k+3\right)\left(k+4\right)}{4}=R$$

Use the distributive property to multiply k^{2}+3k+2 by k+3 and combine like terms.

$$\frac{\left(k^{3}+6k^{2}+11k+6\right)\left(k+4\right)}{4}=R$$

Use the distributive property to multiply k^{3}+6k^{2}+11k+6 by k+4 and combine like terms.

$$\frac{k^{4}+10k^{3}+35k^{2}+50k+24}{4}=R$$

Divide each term of k^{4}+10k^{3}+35k^{2}+50k+24 by 4 to get \frac{1}{4}k^{4}+\frac{5}{2}k^{3}+\frac{35}{4}k^{2}+\frac{25}{2}k+6.

$$\frac{1}{4}k^{4}+\frac{5}{2}k^{3}+\frac{35}{4}k^{2}+\frac{25}{2}k+6=R$$

Swap sides so that all variable terms are on the left hand side.

$$R=\frac{1}{4}k^{4}+\frac{5}{2}k^{3}+\frac{35}{4}k^{2}+\frac{25}{2}k+6$$