Try to simplify the following rational expression:

$$\frac{36s^2 - 12st + t^2}{t^2 + 2ts - 48s^2} = \frac{(6s - t)^2}{(\_ + \_)(\_ + \_)}$$

(Hint: Factor $t^2 + 2ts - 48s^2$ and simplify the rational expression, assuming that $t^2 + 2ts - 48s^2
eq 0$.) [Page No. 85]

Factor the numerator: $36s^2 - 12st + t^2$

$$= (6s - t)^2 = (t - 6s)^2.$$

Factor the denominator: $t^2 + 2ts - 48s^2$

$$= (t + 8s)(t - 6s),$$

since $8s$ and $-6s$ add to $2s$ and multiply to $-48s^2$.

Substituting these factors,

$$\frac{36s^2 - 12st + t^2}{t^2 + 2ts - 48s^2} = \frac{(t - 6s)^2}{(t + 8s)(t - 6s)}$$

Cancelling the common factor $(t - 6s)$, $\frac{t - 6s}{t + 8s}$

Hence, the blanks are $(t + 8s)(t - 6s)$.

This simplification is valid provided $t^2 + 2ts - 48s^2
eq 0$.