If a and b are the roots of the equation $$x^2-5x-14=0$$​, then the value of $$a^3 b^2+a^2 b^3$$​ is:

Correct option is B

Given:

a and b are the roots of the equation $$x^2-5x-14=0,$$

Solution:

The quadratic equation can be factorized as:

​$$x ^2 −5x−14$$ = 0

​​$$x ^2 −(7x+2x)−14$$ = 0

​​$$x(x−7)+2(x−7)$$ = 0

​​$$(x+2)(x−7)$$ = 0

​​the roots of the equation are:

x = −2 and x =7.

a = -2 and b = 7

the value of $$a ^3 b ^2 +a ^2 b ^3$$ ​This expression can be factored as:

​$$a ^3 b ^2 +a ^2 b ^3 =a ^2 b ^2 (a+b)$$

Given roots a = -2 and b = 7

​$$a+b=−2+7=5,$$

​​$$ab=(−2)(7)=−14.$$

Substitute into the factored expression

​$$a ^2 b ^2 (a+b)$$

= $$(−14) ^2 ×5$$

= −14×(−14×5)

= −14 × −70

= 980

the value of  $$a ^3 b ^2 +a ^2 b ^3$$  =  980

Thus, correct answer is (b) 980