To find \( f(x) \cdot g(x) \), we first need to substitute the given functions into the expression.
We know: \[ f(x) = -x - 1 \] \[ g(x) = -10x^2 + x - 6 \]
Now, we will compute \( f(x) \cdot g(x) \): \[ f(x) \cdot g(x) = (-x - 1) \cdot (-10x^2 + x - 6) \]
Now let's perform the multiplication by distributing \( -x - 1 \) to each term in \( g(x) \):
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First, multiply \( -x \) with each term in \( g(x) \): \[ -x \cdot (-10x^2) = 10x^3 \] \[ -x \cdot x = -x^2 \] \[ -x \cdot (-6) = 6x \]
So, from this part, we get: \[ 10x^3 - x^2 + 6x \]
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Next, multiply \( -1 \) with each term in \( g(x) \): \[ -1 \cdot (-10x^2) = 10x^2 \] \[ -1 \cdot x = -x \] \[ -1 \cdot (-6) = 6 \]
So, from this part, we get: \[ 10x^2 - x + 6 \]
Now, we combine all the terms obtained from Steps 1 and 2: \[ f(x) \cdot g(x) = (10x^3 - x^2 + 6x) + (10x^2 - x + 6) \]
Combining like terms:
- The \( x^3 \) term: \( 10x^3 \)
- The \( x^2 \) terms: \( -x^2 + 10x^2 = 9x^2 \)
- The \( x \) terms: \( 6x - x = 5x \)
- The constant term: \( 6 \)
Thus, the resulting polynomial \( f \cdot g \) is: \[ f(x) \cdot g(x) = 10x^3 + 9x^2 + 5x + 6 \]
Therefore, the final answer is: \[ \boxed{10x^3 + 9x^2 + 5x + 6} \]