To factor the polynomial 2p^2+7p-6, we need to find two numbers whose product is -12 (the product of the coefficients of the first and last terms) and whose sum is 7 (the coefficient of the middle term). The numbers are 3 and 4, so we can write:
2p^2+7p-6 = 2p^2+3p+4p-6
Now we can factor by grouping:
= (2p^2+3p) + (4p-6)
= p(2p+3) + 2(2p+3)
= (p+2)(2p+3)
Therefore, the factored form of 2p^2+7p-6 is (p+2)(2p+3).
To factor the polynomial -5v^2+31v-6, we need to find two numbers whose product is -30 (the product of the coefficients of the first and last terms) and whose sum is 31 (the coefficient of the middle term). The numbers are 30 and 1, so we can write:
-5v^2+31v-6 = -5v^2+30v+v-6
Now we can factor by grouping:
= (-5v^2+30v) + (v-6)
= -5v(v-6) + 1(v-6)
= (v-6)(-5v+1)
Therefore, the factored form of -5v^2+31v-6 is (v-6)(-5v+1).
To factor the polynomial -6v^2-11v-4, we need to find two numbers whose product is -24 (the product of the coefficients of the first and last terms) and whose sum is -11 (the coefficient of the middle term). The numbers are -8 and 3, so we can write:
-6v^2-11v-4 = -6v^2-8v+3v-4
Now we can factor by grouping:
= (-6v^2-8v) + (3v-4)
= -2v(3v+4) + 1(3v-4)
= (3v-4)(-2v+1)
Therefore, the factored form of -6v^2-11v-4 is (3v-4)(-2v+1).