Sure, let's factor the expression step-by-step:
Given expression:
[tex]\[ 6a^4(2a + 7) - 5(2a + 7) \][/tex]
1. Identify the common factor: Notice that both terms in the expression have a common factor of [tex]\(2a + 7\)[/tex].
2. Factor out the common factor: Pull the common factor [tex]\(2a + 7\)[/tex] out in front of a parenthesis. This means we write the expression as a product of [tex]\(2a + 7\)[/tex] and another expression.
[tex]\[ = (2a + 7) \left( 6a^4 - 5 \right) \][/tex]
3. Verify the factors by expanding back:
To check your factorization, you can expand it back out:
[tex]\[ (2a + 7)(6a^4 - 5) \][/tex]
Using the distributive property (FOIL method in this case):
[tex]\[ = (2a + 7) \times 6a^4 + (2a + 7) \times (-5) \][/tex]
[tex]\[ = 6a^4 \times 2a + 6a^4 \times 7 - 5 \times 2a - 5 \times 7 \][/tex]
[tex]\[ = 12a^5 + 42a^4 - 10a - 35 \][/tex]
This verifies that our factorization is correct.
Therefore, the fully factored form of the expression [tex]\(6a^4(2a + 7) - 5(2a + 7)\)[/tex] is:
[tex]\[
(2a + 7)(6a^4 - 5)
\][/tex]
