Answer: To factor the expression 3r^2 - 10r - 8, we look for two binomials that, when multiplied together, give us the original expression. The general form of factoring a quadratic expression ax^2 + bx + c is (mx + n)(px + q). We need to find m, n, p, and q in this case.
Let's start by factoring it step by step:
Multiply the coefficient of the squared term (a) with the constant term (c): 3 * -8 = -24.
Find two numbers that multiply to -24 and add up to the coefficient of the middle term (b), which is -10. The numbers are -12 and +2 since -12 * 2 = -24 and -12 + 2 = -10.
Now rewrite the middle term (-10r) using the two numbers (-12r + 2r).
The expression becomes: 3r^2 - 12r + 2r - 8.
Now, factor by grouping:
Group the first two terms and the last two terms separately:
(3r^2 - 12r) + (2r - 8).
Factor out the greatest common factor from each group:
3r(r - 4) + 2(r - 4).
Notice that both terms now have a common factor, which is (r - 4).
The factored expression is: (3r + 2)(r - 4).