If one root of the equation $$x^2 + Px + 84 = 0$$ is $$-21$$, then how much greater is the value of $$P$$ than the other root of the equation?

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**Solution:**

1. Given the quadratic equation:
   $$x^2 + Px + 84 = 0$$

2. One root is given as $$-21$$. Let the other root be $$eta$$.

3. Using the sum of roots formula for quadratic equations:
   $$	ext{Sum of roots} = -P = (-21) + \beta$$

Therefore,
   $$-P = -21 + \beta \implies \beta = -P + 21$$

4. Using the product of roots formula:
   $$	ext{Product of roots} = 84 = (-21) 	imes \beta$$

So,
   $$84 = -21 	imes \beta \implies \beta = \frac{84}{-21} = -4$$

5. From step 3, we have:
   $$\beta = -P + 21$$

Substitute $$\beta = -4$$:
   $$-4 = -P + 21 \implies -P = -4 - 21 = -25 \implies P = 25$$

6. The question asks: "How much greater is the value of $$P$$ than the other root?"

Other root is $$\beta = -4$$

So,
   $$P - \beta = 25 - (-4) = 25 + 4 = 29$$

However, the options do not include 29 as the difference between $$P$$ and the other root.

Re-examining the question, it asks: "the value of $$P$$ is how much more than another root?" This can be interpreted as the difference $$P - \beta$$, which is 29.

But option (e) is 29, which is present.

So the correct answer is 29.

**Answer:** $29$

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