How do you combine like terms in #(6n ^ { 3} - 7+ 6n ^ { 2} ) + ( 4- 7n ^ { 2} - 6n ^ { 4}) + (8n^4 - 8n )#?

Given -

#(6n^3-7+6n^2)+(4-7n^2-6n^4)+(8n^4-8n)#

#6n^3-7+6n^2+4-7n^2-6n^4+8n^4-8n#

#-6n^4+8n^4+6n^3+6n^2-7n^2-8n-7+4#

#2n^4+6n^3-n^2-8n-3#

Use BEDMAS or PEDMAS (use the one you have been taught, they do the same thing) to simplify :

First: B (= brackets), or P (= parentheses).

When removing brackets, multiply each term by the quantity outside the brackets.

If the quantity outside the brackets is 1, the 1 is not needed and normally not shown.

Five examples
# + (x + 1) = +1(x + 1) = (1*x) + (1*1) = x+1#

# - (x + 1) = -1(x + 1) = (-1*x ) + (-1*1) = -x-1#
(multiplying by a negative changes the sign)

#a(x + 1) = a*x + a*1 = ax + a#
#-a(x + 1) = -a*x - a*1 = -ax - a#
#-a(x - 1) = -a*x + a*1 = -ax + a#

In this question:

#(6n ^ { 3} - 7+ 6n ^ { 2} ) + ( 4- 7n ^ { 2} - 6n ^ { 4}) + (8n^4 - 8n )#
All these brackets have + in front of them, so we can remove the brackets and all terms are unchanged.

#6n ^ { 3} - 7+ 6n ^ { 2} + 4- 7n ^ { 2} - 6n ^ { 4} + 8n^4 - 8n #

Rearranging the expression so like terms are together (and in order):

#+ 8n^4 - 6n ^ { 4} + 6n ^ { 3} - 7n ^ { 2} + 6n ^ { 2} - 8n - 7 + 4#

Second: E = exponents. Evaluate exponents where possible.
In this expression there are no exponents that can be evaluated or simplified.

Third: D = divide and M = multiply. There are no divisions or multiplications here.

Fourth: A = add and S = subtract. Only like terms can be added or subtracted. Different powers of an unknown are not like terms.

Adding/subtracting like terms simplifies the expression:

#8n^4 - 6n ^ { 4} + 6n ^ { 3} - 7n ^ { 2} + 6n ^ { 2} - 8n - 7 + 4#

# 2n^4 + 6n ^ { 3} - n ^ { 2} - 8n - 3#

The key realization is that we can combine terms with the same degree.

Paying close attention to the sign, we get

#2n^4+6n^3-n^2-8n-3#

Hope this helps!