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Given the polynomial function [tex] f(x) = (2x+7)^3 + 4x^2 + C_1 x + C_2 [/tex], find the constants [tex] C_1 [/tex] and [tex] C_2 [/tex] such that:

1. [tex] f(-4) = 84 [/tex]

2. [tex] f(-3) = 52 [/tex]

Answer :

To solve the problem, we need to find the values of [tex]\( C_1 \)[/tex] and [tex]\( C_2 \)[/tex] in the function [tex]\( f(x) = (2x + 7)^3 + 4x^2 + C_1x + C_2 \)[/tex] using the information that [tex]\( f(-4) = 84 \)[/tex] and [tex]\( f(-3) = 52 \)[/tex].

### Step-by-Step Solution:

1. Substitute [tex]\( x = -4 \)[/tex] into the function:

We know that:
[tex]\[
f(-4) = (2(-4) + 7)^3 + 4(-4)^2 + C_1(-4) + C_2 = 84
\][/tex]

Simplify:
[tex]\[
= (-8 + 7)^3 + 4 \times 16 - 4C_1 + C_2 = 84
\][/tex]
[tex]\[
= (-1)^3 + 64 - 4C_1 + C_2 = 84
\][/tex]
[tex]\[
= -1 + 64 - 4C_1 + C_2 = 84
\][/tex]
[tex]\[
= 63 - 4C_1 + C_2 = 84
\][/tex]

Rearrange to form equation (1):
[tex]\[
-4C_1 + C_2 = 21
\][/tex]

2. Substitute [tex]\( x = -3 \)[/tex] into the function:

We know that:
[tex]\[
f(-3) = (2(-3) + 7)^3 + 4(-3)^2 + C_1(-3) + C_2 = 52
\][/tex]

Simplify:
[tex]\[
= (-6 + 7)^3 + 4 \times 9 - 3C_1 + C_2 = 52
\][/tex]
[tex]\[
= (1)^3 + 36 - 3C_1 + C_2 = 52
\][/tex]
[tex]\[
= 1 + 36 - 3C_1 + C_2 = 52
\][/tex]
[tex]\[
= 37 - 3C_1 + C_2 = 52
\][/tex]

Rearrange to form equation (2):
[tex]\[
-3C_1 + C_2 = 15
\][/tex]

3. Solve the system of equations:

Now, solve the two equations:
- Equation (1): [tex]\(-4C_1 + C_2 = 21\)[/tex]
- Equation (2): [tex]\(-3C_1 + C_2 = 15\)[/tex]

Subtract equation (2) from equation (1) to eliminate [tex]\( C_2 \)[/tex]:

[tex]\[
(-4C_1 + C_2) - (-3C_1 + C_2) = 21 - 15
\][/tex]
[tex]\[
-4C_1 + C_2 + 3C_1 - C_2 = 6
\][/tex]
[tex]\[
-C_1 = 6
\][/tex]
[tex]\[
C_1 = -6
\][/tex]

Substitute [tex]\( C_1 = -6 \)[/tex] back into equation (2):
[tex]\[
-3(-6) + C_2 = 15
\][/tex]
[tex]\[
18 + C_2 = 15
\][/tex]
[tex]\[
C_2 = 15 - 18
\][/tex]
[tex]\[
C_2 = -3
\][/tex]

### Conclusion:
The values for the constants in the function are [tex]\( C_1 = -6 \)[/tex] and [tex]\( C_2 = -3 \)[/tex].

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