I need to factor these two polynimials :9x^2-16 anda^3-64

The polynomials, 9x^{2 }- 16 and a^{3 }- 64 are the differences of two perfect squares and two perfect cubes, respectively.

Let's begin by recognizing the factoring pattern for the difference of two perfect squares:

a^{2} - b^{2} = (a + b)(a - b)

Let's apply this factoring pattern to our polynomial, 9x^{2} - 64, substuting 9x^{2} for a^{2} and 16 for b^{2} which gives us 3x for a and 4 for b.

Plugging in these values for a and b, our factored expression is (3x + 4)(3x - 4).

Now let's consider the factoring pattern for the difference of two perfect cubes:

a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

Let's use this factoring pattern to factor our polynomial, a^{3} - 64, substituting a for a and 4 for b.

Plugging in thesen values for a and b, our factored expression is (a - 4)(a^{2} + 4a + 16).