Factor by grouping:

30x to the third power-42x square+5x-7=

30x^3 - 42x^2 + 5x - 7

(30x^3 + 5x) + (-42x^2 - 7)
5x( 6x^2 + 1) - 7(6x^2 + 1)
(6x^2 + 1) (5x - 7)

To factor by grouping, we need to group the terms in pairs and look for common factors within each pair. Let's factor the given expression step by step:

1. Group the terms: 30x^3 - 42x^2 + 5x - 7
(30x^3 - 42x^2) + (5x - 7)

2. Find the greatest common factor (GCF) of each pair:
For the first pair (30x^3 - 42x^2), the GCF is 6x^2:
6x^2(5x - 7)

For the second pair (5x - 7), there is no common factor.

3. Combine the factored terms:
6x^2(5x - 7) + (5x - 7)

Now, notice that both terms (6x^2) and (5x - 7) have a common factor of (5x - 7). We can combine them by factoring out this common binomial.

4. Factor out the common binomial:
(5x - 7)(6x^2 + 1)

So, the factored form of the expression 30x^3 - 42x^2 + 5x - 7 is (5x - 7)(6x^2 + 1).