To factor the expression [tex]\(20xy - 35x - 12y + 21\)[/tex] into the form [tex]\((5x - A)(4y - B)\)[/tex], we need to find the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. We will proceed step-by-step.
Starting with the given expression:
[tex]\[ 20xy - 35x - 12y + 21 \][/tex]
We want to express this in the factored form:
[tex]\[ (5x - A)(4y - B) \][/tex]
First, we will expand [tex]\((5x - A)(4y - B)\)[/tex] to see what it looks like:
[tex]\[ (5x - A)(4y - B) = 5x \cdot 4y + 5x \cdot (-B) + (-A) \cdot 4y + (-A) \cdot (-B) \][/tex]
[tex]\[ = 20xy - 5Bx - 4Ay + AB \][/tex]
We will match the terms with the expression [tex]\(20xy - 35x - 12y + 21\)[/tex].
From [tex]\(20xy - 5Bx - 4Ay + AB\)[/tex], we must have:
[tex]\[ 20xy - 5Bx - 4Ay + AB = 20xy - 35x - 12y + 21 \][/tex]
By comparing the coefficients of each term, we can set up the following equations:
For the coefficient of [tex]\(x\)[/tex]:
[tex]\[ -5B = -35 \][/tex]
[tex]\[ B = \frac{35}{5} \][/tex]
[tex]\[ B = 7 \][/tex]
For the coefficient of [tex]\(y\)[/tex]:
[tex]\[ -4A = -12 \][/tex]
[tex]\[ A = \frac{12}{4} \][/tex]
[tex]\[ A = 3 \][/tex]
Next, let's verify these values by expanding with [tex]\(A = 3\)[/tex] and [tex]\(B = 7\)[/tex]:
[tex]\[ (5x - 3)(4y - 7) \][/tex]
[tex]\[ = 5x \cdot 4y + 5x \cdot (-7) + (-3) \cdot 4y + (-3) \cdot (-7) \][/tex]
[tex]\[ = 20xy - 35x - 12y + 21 \][/tex]
This matches the original expression exactly.
Therefore, the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
[tex]\[ A = 3 \][/tex]
[tex]\[ B = 7 \][/tex]
So, [tex]\(A = 3\)[/tex] and [tex]\(B = 7\)[/tex] are the positive integers we were looking for.
The answer is:
[tex]\[ \boxed{37} \][/tex]
