Answer with proof:
To factor each quadratic trinomial, we can use the method of finding two numbers that add up to the coefficient of x and multiply to the constant term. Then we can write the trinomial as a product of two binomials with those numbers as the constant terms. For example:
1. x2+7x + 10
The two numbers that add up to 7 and multiply to 10 are 5 and 2. So we can write:
x2+7x + 10 = (x + 5)(x + 2)
2. x2 -4x-60
The two numbers that add up to -4 and multiply to -60 are -10 and 6. So we can write:
x2 -4x-60 = (x - 10)(x + 6)
3. 2x2 + 17x+21
The two numbers that add up to 17 and multiply to 2x21 are 14 and 3. But we also need to divide them by the coefficient of x2, which is 2. So we can write:
2x2 + 17x+21 = 2(x + 7)(x + 1.5)
4. 3x2-x-10
The two numbers that add up to -1 and multiply to 3x-10 are -3 and 2. But we also need to divide them by the coefficient of x2, which is 3. So we can write:
3x2-x-10 = 3(x - 1)(x + 0.33)
5. 2x312x2 - 14x
To factor this expression completely, we need to use more than one method. First, we can use the greatest common factor method to find the common factor of all the terms, which is 2x. Then we can write:
2x312x2 - 14x = 2x(x3 + 6x - 7)
Next, we can use the quadratic trinomial method to factor the expression inside the parentheses. The two numbers that add up to 6 and multiply to -7 are 7 and -1. So we can write:
2x(x3 + 6x - 7) = 2x(x + 7)(x - 1)
Finally, we can use the difference of squares method to factor the expression x + 7, which is a difference of two perfect squares. We can write:
x + 7 = (x + sqrt(7))(x - sqrt(7))
So the final answer is:
2x312x2 - 14x = 2x(x + sqrt(7))(x - sqrt(7))(x - 1)