To find the factors of [tex]\( A - B \)[/tex], we first need to determine [tex]\( A - B \)[/tex] by subtracting the given functions [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given:
[tex]\[ A = 5x^2 - 5x + 7 \][/tex]
[tex]\[ B = 3x^2 - 18x - 13 \][/tex]
We subtract [tex]\( B \)[/tex] from [tex]\( A \)[/tex]:
[tex]\[ A - B = (5x^2 - 5x + 7) - (3x^2 - 18x - 13) \][/tex]
Distribute the negative sign through the polynomial [tex]\( B \)[/tex]:
[tex]\[ A - B = 5x^2 - 5x + 7 - 3x^2 + 18x + 13 \][/tex]
Combine like terms:
[tex]\[ A - B = (5x^2 - 3x^2) + (-5x + 18x) + (7 + 13) \][/tex]
[tex]\[ A - B = 2x^2 + 13x + 20 \][/tex]
Now, we need to factor [tex]\( 2x^2 + 13x + 20 \)[/tex]. To factor this quadratic expression, we look for two numbers that multiply to [tex]\( 2 \times 20 = 40 \)[/tex] and add up to 13. Those numbers are 5 and 8.
Rewrite the middle term (13x) using these two numbers:
[tex]\[ 2x^2 + 13x + 20 = 2x^2 + 5x + 8x + 20 \][/tex]
Group the terms to factor by grouping:
[tex]\[ 2x^2 + 5x + 8x + 20 = (2x^2 + 5x) + (8x + 20) \][/tex]
Factor out the common factors in each group:
[tex]\[ = x(2x + 5) + 4(2x + 5) \][/tex]
We can see that [tex]\( (2x + 5) \)[/tex] is a common factor:
[tex]\[ = (x + 4)(2x + 5) \][/tex]
Thus, the factors of [tex]\( A - B \)[/tex] are:
[tex]\[ \boxed{(x + 4)(2x + 5)} \][/tex]
