If
(
3
𝑥
+
𝑎
)
(
2
𝑥
+
𝑏
)
=
𝑐
𝑥
2
+
𝑑
𝑥
+
𝑒
(3x+a)(2x+b)=cx
2
+dx+e,
(
3
𝑥
+
𝑏
)
(
2
𝑥
+
𝑎
)
=
𝑐
𝑥
2
+
𝑒
𝑥
+
𝑑
(3x+b)(2x+a)=cx
2
+ex+d for all values of
𝑥
x, and
𝑎
,
𝑏
a,b are both nonzero, what is the value of
𝑒
+
𝑑
e+d?

Given the equations:
\[
(3x + a)(2x + b) = cx^2 + dx + e
\]
\[
(3x + b)(2x + a) = cx^2 + ex + d
\]

First, let's expand both equations.

For \((3x + a)(2x + b)\):
\[
(3x + a)(2x + b) = 6x^2 + 3bx + 2ax + ab
\]
Combining like terms:
\[
= 6x^2 + (3b + 2a)x + ab
\]
Thus:
\[
cx^2 + dx + e = 6x^2 + (3b + 2a)x + ab
\]

So, we have:
\[
c = 6, \quad d = 3b + 2a, \quad e = ab
\]

Next, for \((3x + b)(2x + a)\):
\[
(3x + b)(2x + a) = 6x^2 + 3ax + 2bx + ab
\]
Combining like terms:
\[
= 6x^2 + (3a + 2b)x + ab
\]
Thus:
\[
cx^2 + ex + d = 6x^2 + (3a + 2b)x + ab
\]

So, we have:
\[
c = 6, \quad e = 3a + 2b, \quad d = ab
\]

We now need to determine \(e + d\). From the equations above:
\[
d = ab, \quad e = 3a + 2b
\]

Thus:
\[
e + d = (3a + 2b) + ab
\]

Therefore, the value of \(e + d\) is:
\[
\boxed{3a + 2b + ab}
\]

Please let me know if you have any questions or need further details. Here are some related questions you might find interesting:

1. How would the solution change if \(a\) or \(b\) were zero?
2. What methods can be used to solve for \(a\) and \(b\) given specific values for \(c\), \(d\), and \(e\)?
3. How do the quadratic coefficients affect the roots of the equation?
4. Can the quadratic formula be applied here? If so, how?
5. What are the implications if \(a\) and \(b\) are equal?
6. What is the significance of the coefficients in determining the parabola's vertex?
7. How do we find the sum of the roots of the given quadratic equation?
8. What happens to the product of the roots given the coefficients \(c\), \(d\), and \(e\)?

**Tip:** When working with quadratic equations, always ensure to align coefficients properly when comparing or combining equations.

Explore how to solve quadratic equations involving coefficient relationships like (3x + a)(2x + b) = cx^2 + dx + e and (3x + b)(2x + a) = cx^2 + ex + d. Understand the step-by-step process and implications of coefficient values on solutions.

Math Problem Statement

If ( 3 𝑥 + 𝑎 ) ( 2 𝑥 + 𝑏 )

𝑐 𝑥 2 + 𝑑 𝑥 + 𝑒 (3x+a)(2x+b)=cx 2 +dx+e, ( 3 𝑥 + 𝑏 ) ( 2 𝑥 + 𝑎 )

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