### Home > AC > Chapter 5 > Lesson 5.1.3 > Problem5-27

5-27.

Use a generic rectangle to multiply the following expressions. Write each solution both as a sum and as a product.

1.  $(2x+5)(x+6)$

Draw a generic rectangle and label the length and width.

 Generic rectangle top left edge$2x$ top right edge$+5$ left edge top$x$ interior top left blank interior top right blank left edge bottom$+6$ interior bottom left blank interior bottomright blank

Fill in the grid by multiplying the length and width of each box.

 Generic rectangle top left edge$2x$ top right edge$+5$ left edge top$x$ interior top left$2x^2$ interior top right$5x$ left edge bottom$+6$ interior bottom left$12x$ interior bottomright $30$

$(2x+5)(x+6)=2x^2+17x+30$

Write an equation showing the product is the sum of the parts.
Area as a sum: $2x^2+12x+5x+30=2x^2+17x+30$
Area as a product: $(2x+5)(x+6)$

1.  $(m−3)(3m+5)$

See the help for part (a).

 Generic rectangle top left edge$m$ top right edge$-3$ left edge top$3m$ interior top left $3m^2$ interior top right$-9m$ left edge bottom$+5$ interior bottom left$5m$ interior bottomright $-15$

Area as product: $(m−3)(3m+5)$
Area as sum: $3m^2−4m−15$

1.  $(12x+1)(x^2−5)$

See the help for part (a).

 Generic rectangle top left edge$\color{red}{12x}$ top right edge$\color{red}{+1}$ left edge top$\color{red}{x^2}$ interior top left$\color{red}{12x^3}$ interior top right$\color{red}{x^2}$ left edge bottom$\color{red}{-5}$ interior bottom  left$\color{red}{-60x}$ interior bottomright $\color{red}{-5}$
1.  $(3−5y)(2+y)$

See the help for part (a).