Explanation:
Step 1: Multiply the coefficient of the quadratic term (2) by the constant term (-42). The result is -84.
Step 2: Find two numbers that multiply to -84 and add up to the coefficient of the linear term (-5). In this case, the numbers are -12 and 7, because -12 * 7 = -84 and -12 + 7 = -5.
Step 3: Rewrite the middle term (-5cd) using the two numbers found in Step 2. We get -12cd + 7cd.
Now we can rewrite the original expression as:
2c^2 - 12cd + 7cd - 42d^2
Step 4: Group the terms into two pairs:
(c^2 - 6cd) + (7cd - 42d^2)
Step 5: Factor out the greatest common factor (GCF) from each pair:
c(c - 6d) + 7d(c - 6d)
Step 6: Notice that both terms now have a common factor of (c - 6d). Factor it out:
(c - 6d)(c + 7d)
Therefore, the factored form of the quadratic expression 2c^2 - 5cd - 42d^2 is (c - 6d)(c + 7d).