Sure, let's solve the given question step-by-step in detail.
### Part (a): Factor the Expression [tex]\(4x(x-5) - 3(x-5)\)[/tex]
1. Given Expression: [tex]\(4x(x-5) - 3(x-5)\)[/tex]
2. Identify Common Factors:
Notice that [tex]\((x-5)\)[/tex] is a common factor in both terms of the expression.
[tex]\[
4x(x-5) - 3(x-5) = (x-5)(4x) - 3(x-5)
\][/tex]
3. Factor Out the Common Term:
Since [tex]\((x-5)\)[/tex] is a common factor, we can factor it out from both terms:
[tex]\[
(x-5)(4x) - (x-5)(3) = (x-5)(4x - 3)
\][/tex]
4. Rewrite in Factored Form:
The expression can now be written as:
[tex]\[
4x(x-5) - 3(x-5) = (x-5)(4x - 3)
\][/tex]
From this detailed step-by-step solution, we find that the expression [tex]\(4x(x-5) - 3(x-5)\)[/tex] factors to [tex]\((x-5)(4x - 3)\)[/tex].
### Multiple Choice Answer for Part (a):
- [tex]\( (x-5)(4x+3) \)[/tex]
- [tex]\( (x+5)(4x-3) \)[/tex]
- [tex]\( (x-5)(4x-3) \)[/tex] ⟸ This is the correct answer.
### Part (b): Factor the Expression [tex]\(12x^2 - 6x + 14x - 7\)[/tex]
1. Given Expression: [tex]\(12x^2 - 6x + 14x - 7\)[/tex]
2. Combine Like Terms: First, combine the like terms:
[tex]\[
12x^2 - 6x + 14x - 7 = 12x^2 + 8x - 7
\][/tex]
3. Factor by Grouping:
Let's use the method of factoring by grouping:
Group the terms to make it easier to factor:
[tex]\[
12x^2 + 8x - 7 = (12x^2 + 8x) - 7
\][/tex]
4. Identify Common Factors in Grouped Terms:
Factor out the common factor from each group:
[tex]\(4x\)[/tex] is a common factor in the first group:
[tex]\[
12x^2 + 8x = 4x(3x + 2)
\][/tex]
Since [tex]\(-7\)[/tex] does not fit any obvious simplification, we factor the expression as:
[tex]\[
4x(3x + 2) - 7
\][/tex]
Unfortunately, since there is no other term to group with [tex]\(-7\)[/tex], we cannot factor this expression any further simplification to binomials directly. Thus, [tex]\(12x^2 + 8x - 7\)[/tex] is already in its simplest form as the polynomial doesn't factor nicely into binomials using integers.
### Summary
So, the factors of [tex]\(4 x(x-5) - 3(x-5)\)[/tex] are [tex]\((x-5)(4x - 3)\)[/tex] which corresponds to option:
- [tex]\( (x-5)(4x-3) \)[/tex]
This concludes the step-by-step solution.
