Question

Factorize:
4x⁴ + 7x² − 2

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ 4x^4+7x^2-2 \\ =4{\left(x^2\right)}^2+7x^2-2 \\ \mathrm{Let}: \\ {x}^2=a \\ \therefore 4x^4+7x^2-2=4a^2+7a-2 \\ \mathrm{Splitting}7\mathrm{such}\mathrm{that}\mathrm{the}\mathrm{sum}\mathrm{is}7\mathrm{and}\mathrm{the}\mathrm{product}\mathrm{is}4\times \left(-2\right)=-8. \\ \mathrm{Clearly},\left(8\right)+\left(-1\right)=7\mathrm{and}8\times \left(-1\right)=-8. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \therefore 4a^2+7a-2=4a^2+8a-a-2 \\ =4a(a+2)-1(a+2) \\ =(a+2)(4a-1) \end{gathered}$$
Substituting $a=x^2$, we get:
$$\begin{gathered} 4x^4+7x^2-2=\left(x^2+2\right)\left(4x^2-1\right) \\ =\left(x^2+2\right)\left[{\left(2x\right)}^2-{\left(1\right)}^2\right] \\ =\left(x^2+2\right)\left(2x-1\right)\left(2x+1\right) \end{gathered}$$