Apply the Factor Theorem.

According to the Factor Theorem, $(x-1)$ is a factor of $P(x)$ if and only if $P(1) = 0$.

Since $P(1) = 4 \neq 0$, $(x-1)$ is not a factor of $P(x)$.
The answer is $\boxed{\text{No, } x-1 \text{ is not a factor}}$.

7. Given that $x+2$ is a factor of the polynomial $P(x) = x^3 + x^2 - 4x - 4$, find the complete factorization of $P(x)$.

Since $(x+2)$ is a factor, we divide by $x = -2$.

-2 | 1   1   -4   -4
   |     -2    2    4
   ------------------
     1  -1   -2    0

The quotient is $x^2 - x - 2$ and the remainder is 0.

The quadratic quotient is $x^2 - x - 2$.
We need two numbers that multiply to -2 and add to -1. These numbers are -2 and 1.

The complete factorization is $\boxed{P(x) = (x+2)(x-2)(x+1)}$.

8. Explain why the following expression is not a polynomial: $f(x) = \frac{3}{x^2} + x$. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The term $\frac{3}{x^2}$ can be written as $3x^{-2}$. Since the exponent of $x$ in this term is $-2$, which is a negative integer, the expression $f(x)$ does not meet the definition of a polynomial.

9. Find the remainder when $P(x) = 3x^4 - 2x^3 + 5x - 7$ is divided by $x+1$.

According to the Remainder Theorem, the remainder when $P(x)$ is divided by $(x+1)$ is $P(-1)$.

The remainder is $\boxed{-7}$.

10. Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. Suppose that $P(1)=6$, $P(2)=15$, $P(3)=28$. Find the values of $a, b$, and $c$. Hence determine the polynomial $P(x)$.

For $P(1)=6$:

(1) $a + b + c = 5$

For $P(2)=15$:

(2) $4a + 2b + c = 7$

For $P(3)=28$:

(3) $9a + 3b + c = 1$

Subtract (1) from (2):

(4) $3a + b = 2$

Subtract (2) from (3):

(5) $5a + b = -6$

Subtract (4) from (5):

Substitute $a = -4$ into (4):

Substitute $a = -4$ and $b = 14$ into (1):

Substitute the values $a = -4$, $b = 14$, and $c = -5$ into $P(x) = x^3 + ax^2 + bx + c$.

The values are $\boxed{a = -4, b = 14, c = -5}$.
The polynomial is $\boxed{P(x) = x^3 - 4x^2 + 14x - 5}$.