The expression 4x³ - bx² + x - c leaves remainders 0 and 30 when divided by x + 1 and 2x - 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.

x + 1 = 0 ⇒ x = -1.

Since, x + 1 is a factor of 4x³ - bx² + x - c. Hence, on substituting x = -1 in above expression, remainder = 0.

⇒ 4(-1)³ - b(-1)² + (-1) - c = 0

⇒ -4 - b - 1 - c = 0

⇒ c = -5 - b …….(i)

Given, on dividing 4x³ - bx² + x - c by (2x - 3) we get remainder = 30.

∴ On substituting x = $\dfrac{3}{2}$ in 4x³ - bx² + x - c, value = 30.

$\Rightarrow 4\Big(\dfrac{3}{2}\Big)^3 - b\Big(\dfrac{3}{2}\Big)^2 + \Big(\dfrac{3}{2}\Big) - c = 30 \\[1em] \Rightarrow 4\Big(\dfrac{27}{8}\Big) - b\Big(\dfrac{9}{4}\Big) + \Big(\dfrac{3}{2}\Big) - c = 30 \\[1em] \Rightarrow \dfrac{27}{2} - \dfrac{9b}{4} + \dfrac{3}{2} - 30 = c \\[1em] \Rightarrow \dfrac{54 - 9b + 6 - 120}{4} = c \\[1em] \Rightarrow \dfrac{-9b - 60}{4} = c ……..(ii)$

From (i) and (ii) we get,

$\Rightarrow -5 - b = \dfrac{-9b - 60}{4} \\[1em] \Rightarrow -20 - 4b = -9b - 60 \\[1em] \Rightarrow -20 + 60 = -9b + 4b \\[1em] \Rightarrow -5b = 40 \\[1em] \Rightarrow b = -8.$

Substituting value of b = -8 in (i) we get,

c = -5 - b = -5 - (-8) = 3.

Substituting b = -8 and c = 3 in 4x³ - bx² + x - c we get,

Expression = 4x³ + 8x² + x - 3

Dividing, 4x³ + 8x² + x - 3 by (x + 1),

$\begin{array}{l} \phantom{x + 1)}{4x^2 + 4x - 3} \ x + 1\overline{\smash{\big)}4x^3 + 8x^2 + x - 3} \ \phantom{x + 1}\underline{\underset{-}{}4x^3 \underset{-}{+} 4x^2} \ \phantom{{x + 1}3x^3+}4x^2 + x \ \phantom{{x + 1}3x^3\enspace\space}\underline{\underset{-}{}4x^2 \underset{-}{+} 4x} \ \phantom{{x + 1}{3x^3+1}{+2}}-3x - 3 \ \phantom{{x + 1}{2x^3+}{+2x^2}{4}}\underline{\underset{+}{-}3x \underset{+}{-} 3} \ \phantom{{x + 1}{2x^3+}{+2x^2-}{4x}}\times \end{array}$

we get, quotient = 4x² + 4x - 3.

Factorising 4x² + 4x - 3,

= 4x² + 6x - 2x - 3

= 2x(2x + 3) - 1(2x + 3)

= (2x - 1)(2x + 3).

Hence, b = -8, c = 3 and x³ + 8x² + x - 3 = (x + 1)(2x - 1)(2x + 3).