We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} t\left(2t^3-t+3\right)&=\\[5pt] t\cdot \left(2t^3\right)-t\cdot t+t\cdot 3&=\\[5pt] 2t^{1+3} - t^{1+1} +3t&=\\[5pt] 2t^4-t^2+3t \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} 25y\left(y^4 - 2y^2 + 10\right) &=\\[5pt] 25y\cdot y^4 + 25y\cdot(- 2y^2) + 25y\cdot 10&=\\[5pt] 25y^{1+4} - 50y^{1+2} + 250y &=\\[5pt] \bbox[4pt,border: 1px solid lightgray]{25y^5-50y^3+250y} \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} -t\left(7t+4\right)&=\\[5pt] (-t) \left(7t\right)+ (-t) (4)&=\\[5pt] ((-1)\cdot 7) (t\cdot t) + ((-1)\cdot 4)(t)&=\\[5pt] (-7) t^{1+1} + (- 4)t&=\\[5pt] -7t^2-4t \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} -4x\left(2x^3-3x\right)&=\\[5pt] (-4x) \left(2x^3\right)+ (-4x) (-3x)&=\\[5pt] ((-4) \cdot 2) (x \cdot x^{3}) + ((-4) \cdot (-3))( x \cdot x)&=\\[5pt] (-8) x^{1+3} + (12) x^{1+1}&=\\[5pt] -8x^4+12x^2 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} -5x\left(x^6 - 2x + 4\right) &=\\[5pt] \left(-5x\right) x^6 + \left( -5x\right) (-2x) + \left(-5x\right) 4 &=\\[5pt] (-5)(x \cdot x^{6}) + ((-5) \cdot (-2))(x \cdot x) + ((-5) \cdot 4) ( x) & = \\[5pt] (-5) \cdot x^{1+6} + (10) x^{1+1} + (-20) x & = \\[5pt] \bbox[4pt,border: 1px solid lightgray]{-5x^7 + 10x^2 -20x} \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} -3r^5 \left(r^4-3r^3-2r\right)&=\\[5pt] (-3r^5) \left(r^4\right) + \left(-3r^5\right) \left(-3r^3\right) + \left(-3r^5\right) (-2r)&=\\[5pt] (-3) (r^5\cdot r^{4}) + ((-3) \cdot (-3))(r^5 \cdot r^{3}) + ((-3) \cdot( -2))(r^5 \cdot r)&=\\[5pt] (-3) r^{5+4} + (9) r^{5+3} + (6)r^{5+1}&=\\[5pt] -3r^{9}+9r^8+6r^6 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} 4x^2\left(\dfrac{1}{8}-\dfrac{x}{2}\right) &=\\[5pt] (4x^2) \cdot \left(\dfrac{1}{8}\right) + (4x^2) \cdot \left( -\dfrac{x}{2} \right) &=\\[5pt] \left(4\cdot \dfrac{1}{8}\right) x^2+\left( 4\cdot \left(-\dfrac{1}{2}\right) \right) (x^{1+2}) &=\\[5pt] \left(\dfrac{1}{2} \right) x^2 +(- 2)x^3 &=\\[5pt] \dfrac{x^2}{2}-2x^3 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} 2b^3 \left(-\dfrac b4 + b^3 - \dfrac 13 \right) &=\\[5pt] 2b^3 \cdot \left(-\dfrac b4\right) + 2b^3 \cdot b^3 + 2b^3 \cdot \left(-\dfrac 13\right) & = \\[5pt] %\left(2 \cdot \left(-\dfrac 14\right)\right) (b^3 \cdot b) + %2 (b^3 \cdot b^3) + %\left(2 \cdot \left(-\dfrac 13\right) \right)b^3 & = \\[5pt] \left(-\dfrac 12\right) (b^{3+1})+ 2 (b^{3+3}) + \left( - \dfrac 23\right) b^3 & = \\[5pt] -\dfrac 12 b^4 + 2 b^6 - \dfrac 23 b^3 & = \\[5pt] \bbox[4pt,border: 1px solid lightgray]{ 2 b^6 -\dfrac 12 b^4} - \bbox[4pt,border: 1px solid lightgray]{\dfrac 23}b^3 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} (3x)^2 \left(x^3 - \dfrac 15 x \right) &=\\[5pt] 3^2 x^2 \left(x^3 - \dfrac 15 x \right) &=\\[5pt] 9x^2 \left(x^3 - \dfrac 15 x \right) &=\\[5pt] (9x^2) ( x^3) + (9x^2) \left( - \dfrac 15 x \right) &=\\[5pt] (9) (x^2\cdot x^3) +\left( 9 \cdot \left( - \dfrac 15 \right) \right)( x^2\cdot x) &=\\[5pt] 9 (x^{2+3}) +\left( -\dfrac 95 \right) (x^{2+1}) &=\\[5pt] \bbox[4pt,border: 1px solid lightgray]{9}x^5 - \bbox[4pt,border: 1px solid lightgray]{\dfrac 95 x^3} \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} -p^4\left(-7p^5q^2+3p^2q^5\right)&=\\[5pt] (-p^4) \left(-7p^5q^2\right)+\left(-p^4\right) \left(3p^2q^5\right)&=\\[5pt] (\, (-1)\cdot (-7)\,)( p^{4}\cdot p^{5})( q^2)+ (\, (-1)\cdot 3)( p^{4}\cdot p^{2})( q^5)&=\\[5pt] (7) (p^{4+5})(q^2)+(-3) (p^{4+2})(q^5)&=\\[5pt] 7p^{9}q^2-3p^{6}q^5 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} 6 st\left(\dfrac{s^3}{2}-\dfrac{t^3}{3}\right) &=\\[5pt] (6 st) \left(\dfrac{s^3}{2}\right) + (6 st) \left(-\dfrac{t^3}{3}\right) &=\\[5pt] \left(6 \cdot \dfrac{1}{2}\right)(s \cdot s^3)(t) + \left(6 \cdot \left(-\dfrac{1}{3}\right) \right)(s)(t \cdot t^3) & = \\[5pt] (3)(s^{1+3})(t) + (-2)(s)(t^{1+3}) & = \\[5pt] 3s^4t - 2st^4 \end{align*}

We multiply each term inside the parentheses by the term outside the parentheses, simplify using the rules of exponents, and then we add the results together:

\begin{align*} 4h^3\left(3k^2 + 6hk - h^2\right) &=\\[5pt] (4h^3) \left( 3k^2 \right) + (4h^3) \left( 6hk \right) + (4h^3) \left( - h^2 \right)&=\\[5pt] %(4\cdot 3) \left( h^3 \cdot k^2 \right) + (4\cdot 6) (h^3\cdot h)(k ) + (4\cdot(-1)) \left(h^3\cdot h^2 \right)&=\\[5pt] (12) (h^3)( k^2) + (24) (h^{3+1})(k)+(- 4) h^{3+2} &=\\[5pt] \bbox[4pt,border: 1px solid lightgray]{12 h^3 k^2 + 24 h^4 k} - \bbox[4pt,border: 1px solid lightgray]{4}h^5 \end{align*}