Expand and simplify the following expression. A) (2x^2 + 3)(4x + 5) - 8x(x^2 - 2) B) (4x + 5y)(2x...

Question:

Expand and simplify the following expression.

A) {eq}(2x^2 + 3)(4x + 5) - 8x(x^2 - 2) {/eq}

B) {eq}(4x + 5y)(2x - 3y) + 3xy {/eq}

Polynomials and Algebraic Expressions:

Some algebraic expressions that are affected by operators, such as product and sum, can be expanded by applying the pertinent properties or formulas of the product. They can then be simplified by regrouping similar terms and applying algebraic addition rules.

We know that

{eq}\left ( a+b \right )\left ( c+d \right )=ac+ad+bc+bd {/eq}, and

{eq}a\left ( b+c \right )=ab+ac {/eq}

therefore

A) {eq}\begin{align*} ( 2x^2+3 ) ( 4x+5 )-8x(x^2-2)&= 8x^3+10x^2+12x+15-8x^3+16x \\ &= 8x^3-8x^3+10x^2+12x+16x+15\\ &=10x^2+28x+15 \end{align*} {/eq}

B) {eq}\begin{align*} ( 4x+5y ) ( 2x-3y )+3xy &= 8x^2-12xy+10xy-15y^23xy \\ &=8x^2-12xy+10xy-15y^2+3xy \\ &=8x^2-12xy+10xy+3xy-15y^2 \\ &=8x^2+xy-15y^2 \end{align*} {/eq}