If $$a^3 + b^3 = 20$$ and $$a + b = 5$$, then find the value of $$a^4 + b^4$$.

Given, $$a^3 + b^3 = 20$$ ...........(1)

$$a + b = 5$$

$$=$$> $$\left(a+b\right)^3=5^3$$

$$=$$> $$a^3+b^3+3ab\left(a+b\right)=125$$

$$=$$> $$20+3ab\left(5\right)=125$$

$$=$$> $$15ab=105$$

$$=$$> $$ab=\frac{105}{15}$$

$$=$$> $$ab=7$$ ..........................(2)

$$a + b = 5$$

$$=$$> $$\left(a+b\right)^2=5^2$$

$$=$$> $$a^2+b^2+2ab=25$$

$$=$$> $$a^2+b^2+2\left(7\right)=25$$

$$=$$> $$a^2+b^2=25-14$$

$$=$$> $$a^2+b^2=11$$..................(3)

$$\therefore\ $$ $$\left(a^3+b^3\right)\left(a+b\right)=\left(20\right)\left(5\right)$$

$$=$$> $$a^4+a^3b+b^3a+b^4=100$$

$$=$$> $$a^4+b^4+ab\left(a^2+b^2\right)=100$$

$$=$$> $$a^4+b^4+\left(7\right)\left(11\right)=100$$

$$=$$> $$a^4+b^4+77=100$$

$$=$$> $$a^4+b^4=100-77$$

$$=$$> $$a^4+b^4=23$$

Hence, the correct answer is Option B