If $$\tan A$$ and $$\tan B$$ are the roots of the quadratic equation, $$3x^2 - 10x - 25 = 0$$ then the value of $$3\sin^2(A+B) - 10\sin(A+B) \cdot \cos(A+B) - 25\cos^2(A+B)$$ is:

We are told that $$\tan A$$ and $$\tan B$$ are the roots of the quadratic equation $$3x^2-10x-25=0$$. For any quadratic $$ax^2+bx+c=0$$ whose roots are $$\alpha$$ and $$\beta$$, we have the relations

$$\alpha+\beta=-\dfrac{b}{a},\qquad \alpha\beta=\dfrac{c}{a}.$$

Applying this to the given quadratic, we obtain

$$\tan A+\tan B=\dfrac{10}{3},\qquad \tan A\cdot\tan B=-\dfrac{25}{3}.$$

Next we need $$\tan(A+B)$$. Using the tangent-addition formula

$$\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\,\tan B},$$

we substitute the values just found:

$$\tan(A+B)=\dfrac{\dfrac{10}{3}}{1-\!\left(-\dfrac{25}{3}\right)}=\dfrac{\dfrac{10}{3}}{1+\dfrac{25}{3}} =\dfrac{\dfrac{10}{3}}{\dfrac{28}{3}}=\dfrac{10}{28}=\dfrac{5}{14}.$$

Let us denote $$T=\tan(A+B)=\dfrac{5}{14}.$$

The required expression is

$$E=3\sin^2(A+B)-10\sin(A+B)\cos(A+B)-25\cos^2(A+B).$$

To rewrite everything in terms of $$T$$, we recall the identities

$$\sin^2\theta=\dfrac{\tan^2\theta}{1+\tan^2\theta},\qquad \sin\theta\cos\theta=\dfrac{\tan\theta}{1+\tan^2\theta},\qquad \cos^2\theta=\dfrac{1}{1+\tan^2\theta}.$$

Using $$\theta=A+B$$, every term acquires the common denominator $$1+T^2$$:

$$E=\dfrac{3T^2-10T-25}{1+T^2}.$$

Since $$T=\dfrac{5}{14},$$ we first compute

$$T^2=\left(\dfrac{5}{14}\right)^2=\dfrac{25}{196},\qquad 1+T^2=1+\dfrac{25}{196}=\dfrac{196+25}{196}=\dfrac{221}{196}.$$

Now the numerator becomes

$$3T^2-10T-25 =3\!\left(\dfrac{25}{196}\right)-10\!\left(\dfrac{5}{14}\right)-25 =\dfrac{75}{196}-\dfrac{50}{14}-25.$$

To combine, we convert each term to the common denominator 196:

$$\dfrac{50}{14}=\dfrac{50\times14}{14\times14}=\dfrac{700}{196},\qquad 25=\dfrac{25\times196}{196}=\dfrac{4900}{196}.$$

Hence

$$3T^2-10T-25 =\dfrac{75-700-4900}{196} =\dfrac{-5525}{196}.$$

Putting numerator and denominator together, we obtain

$$E=\dfrac{-5525/196}{221/196} =-\dfrac{5525}{221}.$$

Recognising that $$221\times25=5525,$$ we simplify:

$$E=-25.$$

Hence, the correct answer is Option B.