To add the polynomials, combine the like terms.

(3x² - 3x + 1) + (2x² + x + 4)

3x² + 2x² - 3x + x + 1 + 4

Combine the like terms:

5x² - 2x + 5

To factor the polynomial, find two binomials whose product gives the original polynomial:

6x² - 7x - 20

First, find two numbers that multiply to give the product of the leading coefficient (6) and the constant (-20), which is -120.

The two numbers that multiply to give -120 and add to give the middle term coefficient (-7) are -15 and 8.

Now, rewrite the polynomial, replacing the middle term with the sum -15x + 8x:

6x² - 15x + 8x - 20

Factor by grouping:

3x(2x - 5) + 4(2x - 5)

Both expressions in parentheses are the same, so factor it out:

(2x - 5)(3x + 4)

To find the product, use the distributive property (FOIL):

(3x - 2)(x + 4)

Multiply the first terms, outer terms, inner terms, and last terms:

3x(x) + 3x(4) - 2(x) - 2(4)

Simplify the expression:

3x² + 12x - 2x - 8

Combine the like terms:

3x² + 10x - 8

First, factor the numerator and denominator:

(x² - 4x + 4)/(x² - x)

Factor the numerator as a perfect square:

(x - 2)²/(x² - x)

Factor x from the denominator:

(x - 2)²/x(x - 1)

We cannot simplify the expression further. So the simplified expression is:

(x - 2)²/[x(x - 1)]

To multiply the polynomials, use the distributive property (FOIL):

(x - 4)(x + 3)

Multiply the first terms, outer terms, inner terms, and last terms:

x(x) + x(3) - 4(x) - 4(3)

Simplify:

x² + 3x - 4x - 12

Combine the like terms:

x² - x - 12