Productos Notables $$ 2^{a} $$ parte Desarrolla los siguientes binomios de término comi $$ (x+a)(x+b)=x^{2}+(a+b) x+a b $$ $$ (x+a)(x-b)=x^{2}+(a-b) x+a b $$ 1. $$ (2 x+8)(2 x+7) $$ 2. $$ (3 x+4 y)(3 x-2 y) $$ 3. $$ (2 a-4)(2 a+7) $$ 4. $$ \left(4 m^{2}+3 n\right)\left(4 m^{2}-2 n\right) $$ 5. $$ \left(5 x^{2} y+2 x y^{2}\right)\left(5 x^{2} y-3 x y^{2}\right) $$ 6. $$ \left(6 p^{3} q+4 p q^{2}\right)\left(6 p^{3} q-5 p q^{2}\right) $$

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1. Para expandir $$ (2x+8)(2x+7) $$, usamos la propiedad distributiva:

$$
\begin{aligned}
(2x+8)(2x+7) &= 2x(2x+7) + 8(2x+7)\[6mm]
             &= 4x^2 + 14x + 16x + 56\[6mm]
             &= 4x^2 + 30x + 56.
\end{aligned}
$$

2. Para expandir $$ (3x+4y)(3x-2y) $$:

$$
\begin{aligned}
(3x+4y)(3x-2y) &= 3x(3x-2y) + 4y(3x-2y)\[6mm]
                &= 9x^2 - 6xy + 12xy - 8y^2\[6mm]
                &= 9x^2 + 6xy - 8y^2.
\end{aligned}
$$

3. Para expandir $$ (2a-4)(2a+7) $$:

$$
\begin{aligned}
(2a-4)(2a+7) &= 2a(2a+7) - 4(2a+7)\[6mm]
              &= 4a^2 + 14a - 8a - 28\[6mm]
              &= 4a^2 + 6a - 28.
\end{aligned}
$$

4. Para expandir $$ \left(4m^2+3n\right)\left(4m^2-2n\right) $$:

$$
\begin{aligned}
\left(4m^2+3n\right)\left(4m^2-2n\right) &= 4m^2(4m^2-2n) + 3n(4m^2-2n)\[6mm]
                                         &= 16m^4 - 8m^2n + 12m^2n - 6n^2\[6mm]
                                         &= 16m^4 + 4m^2n - 6n^2.
\end{aligned}
$$

5. Para expandir $$ \left(5x^2y+2xy^2\right)\left(5x^2y-3xy^2\right) $$ procedemos multiplicando cada término:

$$
\begin{aligned}
(5x^2y)(5x^2y) &= 25x^4y^2,\[6mm]
(5x^2y)(-3xy^2) &= -15x^3y^3,\[6mm]
(2xy^2)(5x^2y) &= 10x^3y^3,\[6mm]
(2xy^2)(-3xy^2) &= -6x^2y^4.
\end{aligned}
$$

Sumando los términos:

$$
25x^4y^2 - 15x^3y^3 + 10x^3y^3 - 6x^2y^4 = 25x^4y^2 - 5x^3y^3 - 6x^2y^4.
$$

Se puede factorizar un $$ x^2y^2 $$ si se desea:

$$
25x^4y^2 - 5x^3y^3 - 6x^2y^4 = x^2y^2\left(25x^2 - 5xy - 6y^2\right).
$$

6. Para expandir $$ \left(6p^3q+4pq^2\right)\left(6p^3q-5pq^2\right) $$ procedemos igualmente:

$$
\begin{aligned}
(6p^3q)(6p^3q) &= 36p^6q^2,\[6mm]
(6p^3q)(-5pq^2) &= -30p^4q^3,\[6mm]
(4pq^2)(6p^3q) &= 24p^4q^3,\[6mm]
(4pq^2)(-5pq^2) &= -20p^2q^4.
\end{aligned}
$$

Sumando los términos:

$$
36p^6q^2 - 30p^4q^3 + 24p^4q^3 - 20p^2q^4 = 36p^6q^2 - 6p^4q^3 - 20p^2q^4.
$$

De forma factorizada, extrayendo $$ p^2q^2 $$:

$$
36p^6q^2 - 6p^4q^3 - 20p^2q^4 = p^2q^2\left(36p^4 - 6p^2q - 20q^2\right).
$$
1. $$ (2x+8)(2x+7) = 4x^2 + 30x + 56 $$ 2. $$ (3x+4y)(3x-2y) = 9x^2 + 6xy - 8y^2 $$ 3. $$ (2a-4)(2a+7) = 4a^2 + 6a - 28 $$ 4. $$ (4m^2+3n)(4m^2-2n) = 16m^4 + 4m^2n - 6n^2 $$ 5. $$ (5x^2y+2xy^2)(5x^2y-3xy^2) = 25x^4y^2 - 5x^3y^3 - 6x^2y^4 $$ 6. $$ (6p^3q+4pq^2)(6p^3q-5pq^2) = 36p^6q^2 - 6p^4q^3 - 20p^2q^4 $$

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