Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros. $$ q(x)=-4 x^{4}+8 x^{3}-8 x^{2}+3 x+2 $$ If there is more than one possibility, separate them with commas. (a) Possible number(s) of positive real zeros: (b) Possible number(s) of negative real zeros:

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1. Identify the coefficients of  
   $$
   q(x) = -4x^4 + 8x^3 -8x^2 + 3x + 2.
   $$

2. For positive real zeros, list the signs of the coefficients:
   - $$-4x^4$$: negative
   - $$8x^3$$: positive
   - $$-8x^2$$: negative
   - $$3x$$: positive
   - $$+2$$: positive

3. Count the sign changes:
   - From $$-4x^4$$ (negative) to $$8x^3$$ (positive): one change.
   - From $$8x^3$$ (positive) to $$-8x^2$$ (negative): second change.
   - From $$-8x^2$$ (negative) to $$3x$$ (positive): third change.
   - From $$3x$$ (positive) to $$+2$$ (positive): no change.

Therefore, there are 3 sign changes. By Descartes' Rule of Signs, the number of positive real zeros is either 3 or reduced by an even number, giving $$3 - 2 = 1$$.

4. For negative real zeros, compute $$q(-x)$$:
   $$
   q(-x) = -4(-x)^4 + 8(-x)^3 -8(-x)^2 + 3(-x) + 2.
   $$
   Simplify term by term:
   - $$(-x)^4 = x^4$$ so $$-4x^4$$.
   - $$(-x)^3 = -x^3$$ so $$8(-x^3) = -8x^3$$.
   - $$(-x)^2 = x^2$$ so $$-8x^2$$ remains.
   - $$3(-x)= -3x$$.
   - Constant remains $$+2$$.

Thus,
   $$
   q(-x) = -4x^4 - 8x^3 -8x^2 -3x + 2.
   $$

5. List the signs of the coefficients in $$q(-x)$$:
   - $$-4x^4$$: negative
   - $$-8x^3$$: negative
   - $$-8x^2$$: negative
   - $$-3x$$: negative
   - $$+2$$: positive

6. Count the sign changes:
   - The change occurs only from $$-3x$$ (negative) to $$+2$$ (positive): one change.

Thus, by Descartes' Rule of Signs, there is exactly 1 negative real zero.

7. Final answers:
   - (a) Possible number(s) of positive real zeros: $$3, 1$$
   - (b) Possible number(s) of negative real zeros: $$1$$
(a) Possible number(s) of positive real zeros: 3, 1 (b) Possible number(s) of negative real zeros: 1

Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros.

If there is more than one possibility, separate them with commas.
(a) Possible number(s) of positive real zeros:
(b) Possible number(s) of negative real zeros:

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Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros. If there is more than one possibility, separate them with commas. (a) Possible number(s) of positive real zeros: (b) Possible number(s) of negative real zeros: