Problem 113 - Introductory Algebra for College Students

Question

Factor completely. $$10 x^{2}(x+1)-7 x(x+1)-6(x+1)$$

Step-by-Step Solution

Answer

The complete factorization of the expression \(10x^2(x+1) - 7x(x+1) - 6(x+1)\) is \((x+1)(2x + 1)(5x - 6)\).

Step 1: Identify the common factor

Observe the expression and identify the common factor. Here, we can see that each term is multiplied by \((x+1)\) so that is our common factor.

Step 2: Factor out the common factor

We can factor out \((x+1)\) from each term: \(10x^2(x+1) - 7x(x+1) - 6(x+1) = (10x^2 - 7x - 6)(x+1)\).

Step 3: Factor the quadratic expression

Now the problem is reduced to factoring the quadratic expression \(10x^2 - 7x - 6\). This is a trinomial with a non-one leading coefficient. We continue factoring by looking for two numbers that multiply to \(10*(-6) = -60\) and add to \(-7\). The numbers \(-12\) and \(5\) fulfil these criteria: \(10x^2 - 7x - 6 = 10x^2 - 12x + 5x - 6.\)

Step 4: Regroup and factor by grouping

Next, regroup and use factoring-by-grouping: \(10x^2 - 12x + 5x - 6 = 2x(5x - 6) + 1(5x - 6). Now, we have a common factor of \(5x - 6\) to factor out: 2x(5x - 6) + 1(5x - 6) = (2x + 1)(5x - 6).

Step 5: Combine the factors

Finally, we combine our factors to express the original expression as a product of its simplest factors: \(10x^2(x+1) - 7x(x+1) - 6(x+1) = (x+1)(2x + 1)(5x - 6).