Question

The polynomial x^4 + 4x^2 + x + a is denoted as p(x). It is given that x^2 + x + 2 is a factor of p(x). Find the value of a and the other quadratic factor of p(x).
[ DO NOT USE LONG DIVISION]

Answer 1

${x}^{2} + x + 2) {x}^{4} + 4 {x}^{2} + x + a( {x}^{2} - x + 3\\ \: \: \: \: \: \: \: \: \: \: \: \: .........{x}^{4} + {x}^{3} + 2 {x}^{2} ... \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - {x}^{3 } + 2 {x}^{2} + x \\ \: \: \: \: \: \: \: \: \: \: \: ........... .... - {x}^{3} - {x }^{2} - 2x ... \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{2} + 3 x + a \\ \: \: \: \: \: \: \: \: \: \: \: \: .......................3 {x}^{2} +3 x + 6...... \\ finaly \: we \: get \: a = 6 \\ and \: other \: \: factor \: of \: p(x) \: is \\ \: \: \: ({x}^{2} - x + 3 ) \: ans$

I hope it will be answer of your question