To add polynomials, simply group by like terms and perform the indicated operation. Remember, only like-variables can be added to one another:

$\displaystyle \left ( 3z + 18x\right ) + \left ( 15z + 4x\right )$

$\displaystyle \left ( 3z + 15z\right ) + \left ( 18x + 4x\right )$

$\displaystyle 18z + 22x$  is the simplest form of this expression.

$\displaystyle 7x^{2}+3xy-16x^{2}+4y-7xy$

Collect like terms.

$\displaystyle 7x^{2}-16x^{2}+3xy-7xy+4y$

$\displaystyle -9x^{2}-4xy+4y$

In order to simplify this expression, we need to add together like terms. What this means, is that we can only add together different parts of the expression that have the same kind of variable. For example, $\displaystyle 3x^{2}y^{2}$ can only be added to other values that also have $\displaystyle x^{2}y^{2}$. We cannot combine $\displaystyle 3x^{2}y^{2}$ to values that do not have the same exact variable - so, we cannot combine it with $\displaystyle 4x^{2}$, or $\displaystyle 6y$, and so on. Everything about the variables of two terms needs to be exactly the same if we are going to be able to combine them - the only thing that can be different are the coefficients (the numbers in front of the variables).

So, let's look at the expression above, and see if there are any like terms that we can combine.

Starting with $\displaystyle 3x^{2}y^{2}$ - in order to be combined with this term, any other term must have $\displaystyle x^{2}y^{2}$ as their variable. There is one other term in this expression tha thas $\displaystyle x^{2}y^{2}$ as their variable - $\displaystyle 4x^{2}y^{2}$. If two terms hae the exact same variables, the only thing that we need to do in order to combine them is to add their coefficients together:

$\displaystyle 3x^{2}y^{2}+4x^{2}y^{2}= 7x^{2}y^{2}$

Now, let's look at the next term in the expression, $\displaystyle 4x^2$. There are no other terms that have $\displaystyle x^2$ as their variable, so this term will stay the same.

Now, let's look at $\displaystyle 6y$. There is one other term that has $\displaystyle y$ as their vairable: $\displaystyle 2y$. Let's add these two terms together:

$\displaystyle 6y +2y = 8y$

Finally, let's look at our last term which happens to have no variable: $\displaystyle 12$. There are no other terms in the expression that have no variable, so this term will stay the same.

Now, let's add together all of our simplified terms, as well as the terms that could not be simplified:

$\displaystyle 7x^2y^2 + 4x^2 +8y +12$

This is our simplified answer.

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle 3x^2 + 12 x + 12x + 5$

$\displaystyle 3x^2 + 24x + 5$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (10x^2 + 5) + (x(3 + x))$

$\displaystyle 10x^2 + 5 + 3x + x^2$

$\displaystyle 11x^2 + 3x + 5$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all like terms (i.e. same variable, same exponent):

$\displaystyle (2(x + 5)) + (3x^2 + x + 1)$

$\displaystyle 2x +10 +3x^2 + x + 1$

$\displaystyle 3x^2 + 3x + 11$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (y(3 + x)) + (x(y + 1))$

$\displaystyle 3y + xy + xy + x$

$\displaystyle 3y + 2xy + x$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle 2(a +b) +a^2 +3b$

$\displaystyle 2a + 2b +a^2 + 3b$

$\displaystyle a^2 + 2a + 5b$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle (x + y) ^2 +xy$

$\displaystyle x^2 + 2xy +y^2 +xy$

$\displaystyle x^2 + 3xy + y^2$

To solve this problem, first distribute anything through parentheses that needs to be distributed, then combine all terms with like variables (i.e. same variable, same exponent):

$\displaystyle 2ab + a + 2(b +1)$

$\displaystyle 2ab + a + 2b + 2$