Exercise 75 Page 444

To determine which of the given expressions is a factor of the given quadratic trinomial of the form am^2+bm+c, we first want to factor the trinomial completely. To do this, we will rewrite the middle term, bm, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b. 6m^2+7m-5 ⇕ 6m^2+7m+(- 5) We have that a= 6, b=7, and c=- 5. There are now three steps we need to follow in order to rewrite the above expression.

1. Find a c. Since we have that a= 6 and c=- 5, the value of a c is 6* (- 5)=- 30.
2. Find factors of a c. Since ac=- 30, which is negative, we need factors of a c to have opposite signs in order for the product to be negative. Since b=7, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor.

c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result - 1 &30 &-1 + 30 &29 - 2 &15 &-2 + 15 &13 - 3 & 10 & - 3 + 10 &7 - 5 &6 &-5 + 6 &1

3. Rewrite bm as two terms. Now that we know which factors are the ones to be used, we can rewrite bm as two terms. 6m^2+7m-5 ⇕ 6m^2 - 3m+ 10m-5

Finally, we will factor the last expression obtained.

Notice that one of the factors of the given trinomial corresponds to the binomial in option D.