We have to show if (x+2) is a factor of the polynomials \(\displaystyle{P}{\left({x}\right)}={x}^{{4}}-{3}{x}^{{3}}-{16}{x}-{12}\) and \(\displaystyle{Q}{\left({x}\right)}={x}^{{3}}-{3}{x}^{{2}}-{16}{x}-{12}\) or not.

To check the factor we have to see whether (x+2) satisfies the given polynomials.

If it satisfies the given polynomial or gives the answer 0 i.e P(x)=0 or Q(x)=0 we will say that it is a factor of the polynomial, if it doesn't satisfy the given polynomial we will say that it is not a factor of the polynomial.

First, we will put \(\displaystyle{x}+{2}={0}\Rightarrow{x}=-{2}\)

Now, we will substitute x=-2 in the given polynomials P(x) and Q(x) and see if it is a factor of these polynomials or not.

Now,

\(\displaystyle{P}{\left(-{2}\right)}={\left(-{2}\right)}^{{4}}-{3}{\left(-{2}\right)}^{{3}}-{16}{\left(-{2}\right)}-{12}\)

\(\displaystyle{P}{\left(-{2}\right)}={16}+{24}+{32}-{12}\)

\(\displaystyle{P}{\left(-{2}\right)}={60}\)

therefore, \(\displaystyle{P}{\left(-{2}\right)}\ne{0}\)

hence, \(x=-2\) is not a factor of the polynomial P(x)

Now,

\(\displaystyle{Q}{\left(-{2}\right)}={\left(-{2}\right)}^{{3}}-{3}{\left(-{2}\right)}^{{2}}-{16}{\left(-{2}\right)}-{12}\)

\(\displaystyle{Q}{\left(-{2}\right)}=-{8}-{12}+{32}-{12}\)

\(\displaystyle{Q}{\left(-{2}\right)}={0}\)

therefore, \(Q(-2)=0\)

Hence, \(x=-2\) is a factor of the polynomial \(Q(x)\)