The values of a = -10/3, b = 2, c = -2 and d = -61/3
Cubic polynomial
Since the expression is a cubic polynomial, we expand the left side and equate it to the right side
(x + a)(x + 3)(2x+1) = bx³ + cx² + dx - 10.
Expanding the left side of the equation, we have
(x + a)(x + 3)(2x+1) = [x² + (3 + a)x + 3a](2x + 1)
= 2x³ + 2x²(3 + a) + 6ax + x² + (3 + a)x + 3a
Collecting like terms, we have
= 2x³ + [2(3 + a) + 1]x² + (6a + 3 + a)x + 3a = bx³ + cx² + dx - 10.
Equating the coefficients on both sides, we have
b = 2
2(3 + a) + 1 = c ⇒ 6 + 2a + 1 = c ⇒ 7 + 2a = c
6a + 3 + a = d ⇒ 7a + 3 = d
3a = -10 ⇒ a = -10/3
Substituting the value of a into c, we have
c = 7 + 2a
c = 7 + 2(-10/3)
c = 7 - 20/3
c = (14 - 20)/3
c = -6/3
c = -2
Substituting the value of a into d, we have#
d = 7a + 3
d = 7(-10/3) + 3
d = -70/3 + 3
d = (-70 + 9)/3
d = -61/3
So, the values of a = -10/3, b = 2, c = -2 and d = -61/3
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