The quickest way to multiply this out is probably to look at each power of x in descending order from 3 to 0, pick out the choices of terms from each binomial factor that combine to give the power of x and add them together:
x^3 : The only way you can get a term in x^3 is by multiplying all of the leading terms of the binomials together: x xx x xx x. So the coefficient of x^3 is color(blue)(1).
x^2 : The way that you can get a term in x^2 is by choosing one of the binomials to provide a constant term and picking the x term from the other binomials. Hence the coefficient of x^2 in the product of the three binomials is 3+4+5 = color(blue)(12).
x : The way that you can get a term in x is by choosing one of the binomials to provide the x and the other two to provide constant multipliers. Hence the coefficient of x in the product of the three binomials is 4*5+5*3+3*4 = 20+15+12 = color(blue)(47).
1 : The constant term in the product contains no factor of x, so the only way you can get it is by multiplying all of the three constants from the binomials together: 3*4*5 = color(blue)(60)
Hence we find:
y = (x+3)(x+4)(x+5) = x^3+12x^2+47x+60
I have described the process in a fairly lengthy way, but with practice, you can probably do these sums mostly in your head, gathering together the various combinations of terms.
