Factoring Polynomials, algebra homework help

I have provided answers and explanations for the problems below. Each of them are solved using a different formula. Please tell me each formula and help me learn how to solve them. Also, how do I know which equation needs which formula? Please explain.

Factor.
review
4m3 – 3m2 + 16m – 12
You answered:
remember
If a polynomial has four terms, you may be able to factor by grouping. Once the terms are in
standard order, factor out the greatest common factor (GCF) of the first two terms and the
GCF of the second two terms. If the expressions in parentheses match, you can factor by
grouping:
ac + ad + bc + bd
a(c + d) + b(c + d)
(a + b)(c + d)
Factor by grouping.
solve
4m3 – 3m? + 16m – 12
m(4m – 3) + 4(4m – 3)
Factor by grouping; the expressions in parentheses should match
(m? + 4)(4m – 3)
Apply the distributive property
Factor.
3z3 – 3z? + 8z – 8
You answered:
If a polynomial has four terms, you may be able to factor by grouping. Once the terms are in
standard order, factor out the greatest common factor (GCF) of the first two terms and the
GCF of the second two terms. If the expressions in parentheses match, you can factor by
grouping:
ac + ad + bc + bd
a(c + d) + b(c + d)
(a + b)(c + d)
Factor by grouping.
3z3 – 3z? + 8z – 8
3z?(z – 1) + 8(z – 1)
Factor by grouping; the expressions in parentheses should match
(3z+ 8) (2 – 1)
Apply the distributive property
Factor.
3h? + 7h + 4
You answered:
To factor a quadratic of the form ax? + bx + C, write it as
ax? + r1x + r2x + C
where a . c = ri.ro and b = rı + r2. Then factor by grouping.
Look at the given quadratic:
3h? + 7h + 4
The product acis 12, so you need to find a pair of factors with a product of 12. The b term
is 7, so you need to find a pair of factors with a sum of 7. Since the product is positive (12)
and the sum is positive (7), you need both factors to be positive.
Make a list of the possible factor pairs with a product of 12, and then find the one with a sum
of 7.
Factor pairs of a . c = 12 Sum of factor pairs
1.12 = 12
1 + 12 = 13
2.6 = 12
2 + 6 = 8
3.4 = 12
3 + 4 = 7
The factors 3 and 4 have a sum of 7. So, replace the quadratic’s 7h term with 3h and 4h,
and then factor by grouping.
3h2 + 7h + 4
3h2 + 3h + 4h + 4
3h(h + 1) + 4h + 1)
Factor by grouping; the expressions in parentheses should match
(3h + 4)( + 1)