Give possible expressions for the length and breadth of the rectangle having $\mathit{35}{y}^{\mathit{2}}\mathit{+}\mathit{13}y\mathit{-}\mathit{12}$ as its area.

What are the possible expressions for the dimensions of the cuboid whose volume is $3x^2-12x$.

Factorize $\left(x^2+\frac{1}{x^2}\right)-4\left(x+\frac{1}{x}\right)+6$. If the answer is ${\left(px+\frac{q}{x}-r\right)}^2$, then find the value of $p+q+r$.

Factorize $(x+2)\left(x^2+25\right)-10x^2-20x$. If the answer is $(px+q)(rx-s{)}^2$, then find the value of $p+q+r+s$.

Factorize $2a^2+2\sqrt{6}ab+3b^2$. If the answer is $(\sqrt{p}a+\sqrt{q}b{)}^2$, then find the value of $p+q$.

Factorize $a^2+b^2+2(ab+bc+ca)$. If the answer is $(pa+qb)(ra+sb+tc)$, then find the value of $p+q+r+s+t$.

Factorize $4(x-y{)}^2-12(x-y\left)\right(x+y)+9(x+y{)}^2$. If the answer is $(px+qy{)}^2$, then find the value of $p+q$.

Factorize:

$a^2-b^2+2bc-c^2$

If the factorized form of $x{y}^9-y{x}^9$ is $kxy\left(x-y\right)\left(x+y\right)\left(x^m+y^2\right)\left(x^n+y^n\right)$, then find the value of $\left(k+m+n\right)$.