Attn: I assumed -q2 as the last term.
(1.) 6p2 + 5pq - q2 ==> ( )( )
First, look at the variables in the first and last terms and notice that they are perfect squares. With that, put a 'p' in the left hand side of each set of parentheses and a 'q' in the right hand side of each set of parentheses:
( p q )( p q)
Since the q2 is a negative, you know that there has to be a '-' sign in one set of parentheses and a '+' sign in the other set:
( p - q )( p + q )
The first term has a coefficient of 6, which means we have find factors of 6: 1 * 6 and 2 * 3
The middle term has a coefficient of 5, which means that one of the factors of 6 we found above have to equal 5 when subtracted from one another. The only set of factors that will do this is 1 and 6. So, place a 6 in front of the 'p' in the first set of parentheses and you can leave the other one as is since it's coeffiecient is 1:
( 6p - q )( p + q )
We see that this is the answer when we check the factorization works:
(6p - q)(p + q) = 6p(p) + 6p(q) - q(p) - q(q) = 6p2 + 6pq - pq -q2 = 6p2 + 5pq - q2
(2.) 2x3 + 2x2y - 12xy2
Notice that there is a greatest common factor, that being 2x. So we first factor out a 2x from every term in the equation:
2x ( x2 + xy - 6y2) = 2x ( )( )
Now we only need to factor what's inside the parentheses. Place an 'x' on the left hand side of each set of parentheses and a 'y' on the right hand side of each set of parentheses. Since the last term is negative, also place a '+' in on set and a '-' in the other set:
2x ( x + y )( x - y )
Since the coefficient of the last term is a 6 and the middle term has a coefficient of 1, we need to find factors of 6 that will subtract from one another to equal 1. Those factors of 6 are 2 and 3, so we place a 3 in front of the y in the first set of parentheses and a 2 in front of the y in the other set:
2x ( x + 3y )( x - 2y )
You can again check that this is the answer as we did in the problem above.
The first problem 6p2 + 5pq - q2
Multiply the coefficient of first term and the last term 6 * -1 = -6. Now list the factors of -6.
1 * -6 = -6
-1 * 6 = -6
2 * -3 = -6
-3 * 2 = -6
But we need the sum of factors as 5. Since our middle term in the given equation is 5. So, if we sum the factors ( -1 + 6 = 5).
So we'll split the middle term and write
6p2 + 5pq - q2
= 6p2 + 6pq - pq - q2 (since 5pq = +6pq - pq)
Group first two terms and last two terms
= (6p2 + 6pq)+ (- pq - q2)
= 6p(p + q) -q(p + q)
= (p + q)(6p - q) ------> answer
GCF of 2x3, 2x2y, 12xy2 is 2x. So, 2x is common in each term.
= 2x(x2 + xy - 6y2)
Same as above, list the factors of -6. This time the sum of factors should be 1 (coefficient of second term) and if you multiply the factors you should get -6(coeffiecient of third term). So, -2 + 3 = 1 and -2 * 3 = -6.
= 2x(x2 - 2xy + 3xy - 6y2) (since xy = -2xy + 3xy)
= 2x(x(x - 2y) + 3y(x - 2y)) (group first two terms and last two terms)
= 2x((x - 2y)(x + 3y))
= 2x (x - 2y) (x + 3y) ---------> answer
Hope this helps you.