Rational Expression:n(n+1)(n+2)(n+3)

n(n+1)(n+2)(n+3)/2*3 + (n+1)(n+2)(n+3)/2*3*4
= n(n+1)(n+2)(n+3)/2*3*4 + (n+1)(n+2)(n+3)*4/2*3*4
=(n+1)(n+2)(n+3)
=n^2+2n+n+2(n+3)
=n^3+6n^2+11n + 6
and since (n+1)(n+2)(n+3) is on both sides then:
=2n^3+12n^2+22n+12
=2(n^3+6n^2+11n+6)

obvisously unable to simplify any further to make the equation work. Troubleshooting these synthesis problems at the end of the chapter really takes a lot of effort. Providing an answer is fine at least for this equation but finding a way to enable me to proceed in handling future problems would be better in the long term. I'm just learning rational expressions so nothing on deductions or anythinng like that. Please provide steps. Thank you.

n(n+1)(n+2)(n+3)/(2*3) + (n+1)(n+2)(n+3)/(2*3*4)

On 2nd line, why did you multiply denominator of first term by 4, but not the numerator, and why did you multiply numerator of second term by 4, but not the denominator.

To keep equality, then you must multiply both the numerator and denominator of a term by the same value. So 2nd line should be:

= n(n+1)(n+2)(n+3)*4/(2*3*4) + (n+1)(n+2)(n+3)/(2*3*4)

On 3rd line, where do the values in denominator disappear to ?
Where does the factor n in first term disappear to ?
Third line should be

= (4n(n+1)(n+2)(n+3) + (n+1)(n+2)(n+3)) / (2*3*4)

Now we can factor out common factors:

= (n+1)(n+2)(n+3) (4n + 1) / 24

Now multiply 1st and 2nd factors together, and multiply 3rd and 4th factors together

= (n² + 2n + n + 2) (4n² + n + 12n + 3) / 24
= (n² + 3n + 2) (4n² + 13n + 3) / 24

Finally, multiply the two quadratic factors:

= (n² (4n² + 13n + 3) + 3n (4n² + 13n + 3) + 2 (4n² + 13n + 3)) / 24
= (4n⁴ + 13n³ + 3n² + 12n³ + 39n² + 9n + 8n² + 26n + 6) / 24
= (4n⁴ + 25n³ + 50n² + 35n + 6) / 24

P.S. Don't forget parentheses to separate numerator from denominator, or one factor from another factor.