Answer:
6x²(x -7)(x +2)
Step-by-step explanation:
The first step is to look at what you are given. Here, we notice that the greatest common factor of all terms is 6x², so that can be factored out right away.
= 6x²(x² -5x -14)
To factor the quadratic, you look for factors of -14 that have a sum of -5.
-14 = (-14)(1) = (-7)(2) . . . . the only integer factor pairs with a negative sum
The sums of these factor pairs are -13 and -5, so we're interested in the latter. You can use these factors directly in the binomial factors of the quadratic:
= 6x²(x -7)(x +2)
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Additional comments
It is useful to start with the polynomial in standard form (decreasing powers of x), as it is here. That way, if there are any factors of x at the right end of the expression, they will be able to be factored from all terms. Here, the right term is a multiple of x², so x² can be factored from all terms.
Your knowledge of multiplication tables tells you 30 is a multiple of 6. Your knowledge of divisibility rules tells you 84 is even and a multiple of 3 (8+4=12 is a multiple of 3). That means 2·3 = 6 is also a factor of 84. (Most folks learn multiplication tables to 10×10, or maybe 12×12. If the latter, you know 84=7×12, so will be a multiple of 6.) The point here is that you can recognize all of the coefficients as being multiples of 6. Now you know that 6x² is a factor of every term, so is a factor of the expression.
