Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$.
The sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$ is $-2 (1 - 1 + 3 - 5) + 4 (1 + 2) - 3 (1 - 4) = (-2) \cdot (-2) + 4 \cdot 3 - 3 \cdot (-3) = \boxed{25}$.  (The sum of the coefficients in a polynomial can be found by setting the variable equal to 1.)
25
Level 4
Algebra
Problem: Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the sum of the coefficients in the polynomial, we first need to simplify the expression given. The sum of the coefficients of a polynomial can be found by evaluating the polynomial at \( x = 1 \). 

Start by substituting \( x = 1 \) into the polynomial:

1. Substitute \( x = 1 \) into each term of the polynomial:
   - For \( -2(x^7 - x^4 + 3x^2 - 5) \):
     - Calculate \( -2(1^7 - 1^4 + 3 \cdot 1^2 - 5) = -2(1 - 1 + 3 - 5) = -2(-2) = 4 \).
   - For \( 4(x^3 + 2x) \):
     - Calculate \( 4(1^3 + 2 \cdot 1) = 4(1 + 2) = 4 \cdot 3 = 12 \).
   - For \( -3(x^5 - 4) \):
     - Calculate \( -3(1^5 - 4) = -3(1 - 4) = -3(-3) = 9 \).

2. Now, combine the results from each term:
   - The total from the first term is \( 4 \).
   - The total from the second term is \( 12 \).
   - The total from the third term is \( 9 \).

3. Add these values together:
   - \( 4 + 12 + 9 = 25 \).

Thus, the sum of the coefficients in the polynomial is \( 25 \).<|END_OF_LATENT|> The final answer is $25$.