If the product $(3x^2 - 5x + 4)(7 - 2x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$.
One can find this by multiplying the two polynomials and computing the coefficients. Alternatively, we notice that the desired expression is simply the value of $ax^3+bx^2+cx+d$ evaluated at the point $x=2$: $a(2)^3 + b(2)^2 + c(2) + d = 8a + 4b + 2c + d$. Since $(3x^2 - 5x + 4)(7 - 2x)$ and  $ax^3+bx^2+cx+d$ are equivalent expressions, they are equal for all real values of $x$.  In particular, they are equal when $x=2$. So $8a+4b+2c+d=(3 \cdot (2)^2 - 5 \cdot (2) + 4)(7 - 2 \cdot (2)) = 6 \cdot 3 = \boxed{18}.$
18
Level 4
Algebra
Problem: If the product $(3x^2 - 5x + 4)(7 - 2x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coefficients \(a\), \(b\), \(c\), and \(d\) in the polynomial expansion of \((3x^2 - 5x + 4)(7 - 2x)\), we need to perform polynomial multiplication. 

First, distribute each term in the first polynomial by each term in the second polynomial:

1. Multiply \(3x^2\) by \(7\) to get \(21x^2\).
2. Multiply \(3x^2\) by \(-2x\) to get \(-6x^3\).
3. Multiply \(-5x\) by \(7\) to get \(-35x\).
4. Multiply \(-5x\) by \(-2x\) to get \(10x^2\).
5. Multiply \(4\) by \(7\) to get \(28\).
6. Multiply \(4\) by \(-2x\) to get \(-8x\).

Now, combine all these results:

- The \(x^3\) term is \(-6x^3\).
- The \(x^2\) terms are \(21x^2 + 10x^2 = 31x^2\).
- The \(x\) terms are \(-35x - 8x = -43x\).
- The constant term is \(28\).

Thus, the polynomial can be expressed as:

\[
-6x^3 + 31x^2 - 43x + 28
\]

From this, we identify the coefficients:

- \(a = -6\)
- \(b = 31\)
- \(c = -43\)
- \(d = 28\)

Next, we compute \(8a + 4b + 2c + d\):

1. Calculate \(8a = 8 \times -6 = -48\).
2. Calculate \(4b = 4 \times 31 = 124\).
3. Calculate \(2c = 2 \times -43 = -86\).
4. The constant \(d = 28\).

Now, sum these values:

\[
8a + 4b + 2c + d = -48 + 124 - 86 + 28
\]

Calculating step-by-step:

- First, combine \(-48 + 124 = 76\).
- Then, combine \(76 - 86 = -10\).
- Finally, combine \(-10 + 28 = 18\).

Thus, the final answer is \(18\).<|END_OF_LATENT|> The final answer is $18$.