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Use the method of equating coefficients to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:

[tex]\[
(x+4)(ax^2+bx+c)=2x^3+9x^2+3x-4
\][/tex]

A. [tex]\(a=2\)[/tex], [tex]\(b=1\)[/tex], [tex]\(c=1\)[/tex]

B. [tex]\(a=-2\)[/tex], [tex]\(b=1\)[/tex], [tex]\(c=-1\)[/tex]

C. [tex]\(a=2\)[/tex], [tex]\(b=-1\)[/tex], [tex]\(c=-1\)[/tex]

D. [tex]\(a=2\)[/tex], [tex]\(b=1\)[/tex], [tex]\(c=-1\)[/tex]



Answer :

GinnyAnswer
Of course! Let's solve the problem step-by-step by equating coefficients to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:

Given the equation:
[tex]\[ (x + 4)\left(ax^2 + bx + c\right) = 2x^3 + 9x^2 + 3x - 4 \][/tex]

First, let's expand the left-hand side of the equation:
[tex]\[ (x + 4)(ax^2 + bx + c) \][/tex]

Distribute [tex]\(x\)[/tex] and 4 across the terms inside the parentheses:
[tex]\[ = x \cdot (ax^2 + bx + c) + 4 \cdot (ax^2 + bx + c) \][/tex]

[tex]\[ = (ax^3 + bx^2 + cx) + (4ax^2 + 4bx + 4c) \][/tex]

Now, combine like terms:
[tex]\[ = ax^3 + (b + 4a)x^2 + (c + 4b)x + 4c \][/tex]

Equate the expanded left-hand side to the right-hand side of the given equation:
[tex]\[ ax^3 + (b + 4a)x^2 + (c + 4b)x + 4c = 2x^3 + 9x^2 + 3x - 4 \][/tex]

Now, we equate the coefficients of like terms on both sides of the equation:

For [tex]\(x^3\)[/tex] terms:
[tex]\[ a = 2 \][/tex]

For [tex]\(x^2\)[/tex] terms:
[tex]\[ b + 4a = 9 \][/tex]

For [tex]\(x\)[/tex] terms:
[tex]\[ c + 4b = 3 \][/tex]

For the constant term:
[tex]\[ 4c = -4 \][/tex]

Let's solve these equations step-by-step.

1. From the [tex]\(x^3\)[/tex] terms:
[tex]\[ a = 2 \][/tex]

2. Substitute [tex]\(a = 2\)[/tex] into [tex]\(b + 4a = 9\)[/tex]:
[tex]\[ b + 4(2) = 9 \][/tex]
[tex]\[ b + 8 = 9 \][/tex]
[tex]\[ b = 1 \][/tex]

3. Substitute [tex]\(b = 1\)[/tex] into [tex]\(c + 4b = 3\)[/tex]:
[tex]\[ c + 4(1) = 3 \][/tex]
[tex]\[ c + 4 = 3 \][/tex]
[tex]\[ c = -1 \][/tex]

4. Verify [tex]\(c\)[/tex] by substituting into [tex]\(4c = -4\)[/tex]:
[tex]\[ 4(-1) = -4 \][/tex]

Therefore, the values are:
[tex]\[ a = 2, \quad b = 1, \quad c = -1 \][/tex]

The correct answer is:
[tex]\[ \boxed{D. \ a = 2, \ b = 1, \ c = -1} \][/tex]

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