$\displaystyle (3p^{2}+8)+(-p^{2}+5)$

This is not a FOIL problem, as we are adding rather than multiplying the terms in parenteses.

Add like terms to solve.

$\displaystyle 3p^{2}-p^{2}=2p^{2}$

$\displaystyle 8+5=13$

Combining these terms into an expression gives us our answer.

$\displaystyle 2p^{2}+13$

When simplifying polynomials, only combine the variables with like terms.

$\displaystyle 15x^{3}y^{2}$ can be added to $\displaystyle 3x^{3}y^{2}$, giving $\displaystyle 18x^{3}y^{2}$. 

$\displaystyle 4x^{2}$ can be subtracted from $\displaystyle 8x^{2}$ to give $\displaystyle 4x^{2}$.

Combine both of the terms into one expression to find the answer: $\displaystyle 18x^{3}y^{2}+4x^{2}$

$\displaystyle (5c^{2}+5)+(-5c-5)$

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms to solve.

$\displaystyle 5c^{2}$ and $\displaystyle -5c$ have no like terms and cannot be combined with anything.

5 and -5 can be combined however:

$\displaystyle 5-5=0$

This leaves us with $\displaystyle 5c^{2}-5c$.

LCM of $\displaystyle 2,4,6=12$

LCM of $\displaystyle x, x^{2}=x^{2}$

and since $\displaystyle \left ( x^{2} -1 \right )= \left ( x+1 \right )\left ( x-1 \right )$

The LCM  $\displaystyle =12x^{2}\left ( x+1 \right )(x-1)$

First factor the denominators which gives us the following:

$\displaystyle \frac{x}{\left ( x+1 \right )\left ( x-3 \right )} + \frac{1}{\left ( x+1 \right )\left ( x-3 \right )}$

The two rational fractions have a common denominator hence they are like "like fractions".  Hence we get:

$\displaystyle \frac{x+1}{\left ( x+1 \right )\left ( x-3 \right )}$

Simplifying gives us

$\displaystyle \frac{1}{x-3}$

$\displaystyle (2x^{2}-5) + (4x^{3}-3x^{2}+2x-1)$

To simplify you combind like terms: 

$\displaystyle (4x^{3}\boldsymbol{-3x^{2}+2x^{2}} +2x\boldsymbol{-1-5)}$

Answer: 

$\displaystyle 4x^{3}-x^{2}+2x-6$

When combining polynomials, only combine like terms. With the like terms, combine the coefficients. Your answer is $\displaystyle -9x^2y^3z+6m^4n^5$. 

Don't be scared by complex terms! First, we follow our order of operations and multiply the $\displaystyle y$ into the first binomial. Then, we check to see if the variables are alike. If they match perfectly, we can add and subtract their coefficients just like we could if the expression was $\displaystyle 3x + 3x$.

Remember, a variable is always a variable, no matter how complex! In this problem, the terms match after we follow our order of operations! So we just add the coefficients of the matching terms and we get our answer:$\displaystyle 11 a^2 b^2 y+13 x^2 y z^2$

To solve $\displaystyle (x^3+2x-1)+(-2x^3-1)$, identify all the like-terms and regroup to combine the values.

$\displaystyle x^3-2x^3+2x-1-1= -x^3+2x-2$

To add two polynomials together, you combine all like terms.

Combining the $\displaystyle x^3$ terms gives us $\displaystyle 3x^3$

Combining the $\displaystyle x^2$ terms gives us $\displaystyle 3x^2$, since there is only 1 of those terms in the expression it remains the same.

Combining the $\displaystyle x$ terms gives us $\displaystyle x$

and finally combining theconstants gives us $\displaystyle -1$

summing all these together gives us 

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