To determine the correct factors of the quadratic polynomial [tex]\(6x^2 + 37x - 60\)[/tex], we need to verify which pairs of expressions, when multiplied, will give us the original polynomial:
1. Checking [tex]\( (3x - 4) \text{ and } (2x + 15)\)[/tex]:
Multiply these two binomials:
[tex]\[
(3x - 4)(2x + 15)
\][/tex]
Expand the product:
[tex]\[
3x \cdot 2x + 3x \cdot 15 - 4 \cdot 2x - 4 \cdot 15 = 6x^2 + 45x - 8x - 60
\][/tex]
Combine like terms:
[tex]\[
6x^2 + (45x - 8x) - 60 = 6x^2 + 37x - 60
\][/tex]
Since the expanded polynomial [tex]\(6x^2 + 37x - 60\)[/tex] matches the original polynomial, these factors are correct.
2. Checking [tex]\( (3x + 4) \text{ and } (2x - 15)\)[/tex]:
Multiply these two binomials:
[tex]\[
(3x + 4)(2x - 15)
\][/tex]
Expand the product:
[tex]\[
3x \cdot 2x + 3x \cdot (-15) + 4 \cdot 2x + 4 \cdot (-15) = 6x^2 - 45x + 8x - 60
\][/tex]
Combine like terms:
[tex]\[
6x^2 + (-45x + 8x) - 60 = 6x^2 - 37x - 60
\][/tex]
Since the expanded polynomial [tex]\(6x^2 - 37x - 60\)[/tex] does not match the original polynomial, these factors are incorrect.
3. Checking [tex]\( 2(x - 2) \text{ and } 3(x + 5)\)[/tex]:
Multiply these two binomials:
[tex]\[
2(x - 2) \cdot 3(x + 5)
\][/tex]
Expand the product:
[tex]\[
2(x - 2) \cdot 3x + 2(x - 2) \cdot 15 = 2x(3x) + 2x(15) - 4(3x) - 4(15) = 6x^2 + 30x - 8x - 60
\][/tex]
Combine like terms:
[tex]\[
6x^2 + 22x - 60
\][/tex]
Since the expanded polynomial [tex]\(6x^2 + 22x - 60\)[/tex] does not match the original polynomial, these factors are incorrect.
4. Checking [tex]\( 2(x + 2) \text{ and } 3(x - 5)\)[/tex]:
Multiply these two binomials:
[tex]\[
2(x + 2) \cdot 3(x - 5)
\][/tex]
Expand the product:
[tex]\[
2(x + 2) \cdot 3x + 2(x + 2) \cdot (-15) = 2x(3x) - 2x(15) + 4(3x) - 4(15) = 6x^2 - 30x + 12x - 60
\][/tex]
Combine like terms:
[tex]\[
6x^2 - 18x - 60
\][/tex]
Since the expanded polynomial [tex]\(6x^2 - 18x - 60\)[/tex] does not match the original polynomial, these factors are incorrect.
Therefore, the correct factors of the polynomial [tex]\(6x^2 + 37x - 60\)[/tex] are:
[tex]\[ \boxed{3x - 4 \text{ and } 2x + 15} \][/tex]
