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You multiply each term of one binomial by each term of the other binomial. In fact, this works for multiplying any polynomials: multiply each term of one polynomial by each term of the other one. Then add all the terms together.
60 multiplied by 10 is equal to 600. Multiplying two numbers involves adding one of the numbers to itself a certain number of times determined by the other number. In this case, adding 60 to itself 10 times results in a total of 600.
Polynomials have graphs that look like graphs of their leading terms because all other changes to polynomial functions only cause transformations of the leading term's graph.
Numbers go on forever. Any number can be increased by adding 1, or by adding any other number, or by doubling that number, or by multiplying that number by any other positive number greater than 1. All such operations will result in a larger number, and you may perform such operations as many times as desired. ("Infinity" is a concept different from large numbers; it is not itself a number in the class of "real numbers.") You may see the Related Link for more on large numbers.
The purpose is so that on any right triangle, you can figure out how long the length of one of the sides are by multiplying each other side by themselves, adding them together, and then finding the square root of that number.