5. (2*x*^{2} - 3*x* + 5)(*x*^{2} + 4*x* - 6)

This is a multiplication problem. The rule which applies in this situation is the distributive rule, which says that one must multiply the factor outside the parentheses times all the terms in side the parentheses. This would give us

*x*^{2}(2*x*^{2} - 3*x* + 5) + 4*x*(2* x^{2} - *3

This gives us three distributive problems.

(*x*^{2})(2*x*^{2}) - (*x*^{2})(3*x*) + (*x*^{2})(5) + (4*x*)(2*x*^{2}) - (4*x*)(3*x*) + (4*x*)(5) - 6(2*x*^{2}) + 6(3*x*) - 6(5)

Following this chain of ideas, we conclude that to multiply two polynomials, multiply each term in the one polynomial by each term in the other. As a result, the number of terms in the product will be the product of the number of terms.

If you look at these products, you will see that to multiply two terms, multiply the coefficients and add the exponents

2*x*^{4} - 3*x*^{3} + 5*x*^{2} + 8*x*^{3} - 12*x*^{2} + 20*x* - 12*x*^{2} + 18*x* - 30

Now, combine like terms. Let us first rearrange the terms so that the like terms are together.

2*x*^{4} - 3*x*^{3} + 8*x*^{3} + 5*x*^{2}- 12*x*^{2} - 12*x*^{2} + 20*x *+ 18*x* - 30

2*x*^{4}* *+ 5*x*^{3} - 19*x*^{2} + 38*x* + 30

You may find the following vertical format better for keeping track of all the like terms. Write the polynomials down on top of each other with, a line under them, and below the line put the products of the first term in the bottom polynomial by all the terms in the top polynomial. On the next line, put the products of the next term in the bottom polynomial times all the iterms in the top polynomial, lining up the like terms, etc.

and we get the same answer.